09 Exercise 02 - boonnum.files.wordpress.com · ครูบุญนํา เกษี กลุ่มสาระการเรียนรู้คณิตศาสตร์

ใบความรท� 3 การแกระบบสมการเชงเสนโดยใชการแทนคา

การแกระบบสมการเชงเสนโดยการแทนคาตวแปรน�น มหลกการทางาย ๆ คอ จดสมการในรปตวแปรตวแปรหน&ง แลวนาไปแทนคาลงในอกสมการหน&ง แลวดาเนนการแกสมการโดยใชสมบตการเทากนของจานวนจรง ซ& งพอสรปเปนข�นตอนไดดงน�

ใบความรท� 2.2 การแกระบบสมการเชงเสน

โดยใชการแทนคา

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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ใหนกเรยนพจารณาการแกระบบสมการเชงเสนโดยการแทนคาตวแปรตอไปน�

จงแกระบบสมการ 5yx13 =+

5yx8 =+ วธทา 5yx13 =+ สมการท& 1

5x13y +−= นา 13x มาลบท�งสองขาง 55x13x8 =+− แทน y = -13x + 5 ในสมการท& 2 55x5 =+− รปอยางงาย 0x5 =− นา 5 มาลบท�งสองขาง 0x = นา -5 มาหารท�งสองขาง 5yx13 =+ สมการท& 1 5y)0(13 =+ แทน 0x = 5y = รปอยางงาย

5yx13 =+ สมการท& 1 55)0(13 =+ แทน 5y,0x ==

55 = รปอยางงาย (เปนสมการท�เปนจรง)

5yx8 =+ สมการท& 2 55)0(8 =+ แทน 5y,0x ==

55 = รปอยางงาย (เปนสมการท�เปนจรง)

ดงน&น คาตอบของระบบสมการคอ (0, 5)

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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จงแกระบบสมการ 11y7x =+

8y4x =+ วธทา 11y7x =+ สมการท& 1

11y7x +−= นา 7y มาลบท�งสองขาง 8y411y7 =++− แทน 8y4x +−= ในสมการท& 2 811y3 =+− รปอยางงาย 3y3 −=− นา 11 มาลบท�งสองขาง 1y = นา -3 มาหารท�งสองขาง 8y4x =+ สมการท& 2 8)1(4x =+ แทน 1y = 84x =+ รปอยางงาย 4x = นา 4 มาลบท�งสองขาง

11y7x =+ สมการท& 1 11)1(74 =+ แทน 1y,4x ==

1111 = รปอยางงาย (เปนสมการท�เปนจรง)

8y4x =+ สมการท& 2 8)1(44 =+ แทน 1y,4x ==

88= รปอยางงาย (เปนสมการท�เปนจรง)

ดงน&น คาตอบของระบบสมการคอ (4, 1)

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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จงแกระบบสมการ 1yx =+−

2yx2 −=+ วธทา 1yx =+− สมการท& 1 1xy += นา x มาบวกท�งสองขาง 2yx2 −=+ สมการท& 2

21xx2 −=++ แทน 1xy +=

21x3 −=+ รปอยางงาย

3x3 −= นา 1มาลบท�งสองขาง 1x −= นา 3 มาหารท�งสองขาง 1yx =+− สมการท& 1

1y)1( =+−− แทน 1x −=

1y1 =+ รปอยางงาย 111y1 −=−+ นา 1− มาบวกท�งสองขาง 0y = รปอยางงาย

1yx =+− สมการท& 1

10)1( =+−− แทน 1x −= , 0y =

11= รปอยางงาย (เปนสมการท�เปนจรง)

2yx2 −=+ สมการท& 2 20)1(2 −=+− แทน 1x −= และ 0y =

22 −=− รปอยางงาย (เปนสมการท�เปนจรง)

ดงน&น คาตอบของระบบสมการคอ )0,1(−

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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จงแกระบบสมการ 1yx =−

1yx =+− วธทา 1yx =− สมการท& 1 1yx += นา y มาบวกท�งสองขาง 1yx =+− สมการท& 2 1y)1y( =++− แทน 1yx += 1y1y =+−− สมบตการแจกแจง 11=− รปอยางงาย เน&องจาก 11=− เปนสมการท&ไมเปนจรง ดงน&น ระบบสมการน�ไมมคาตอบ

สมการท& 1 1=− yx จดสมการ 1−= xy สมการท& 2 1=+− yx จดสมการ 1+= xy

ถา 21a a= และ 21b b≠ ความชนเทากนแตจดตดบนแกน Y ตางกน กราฟของเสนตรงทงสองจะขนานกนแลว

ดงตวอยางท 4

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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จงแกระบบสมการ 3y2x =+

6y4x2 =+ วธทา 3y2x =+ สมการท& 1 3y2x +−= นา 2y มาลบท�งสองขาง 6y4x2 =+ สมการท& 2

6y4)3y2(2 =++− แทน 3y2x +−=

6y46y4 =++− สมบตการแจกแจง 66 = รปอยางงาย เน&องจาก 66= เปนสมการท&เปนจรง

ดงน&น ระบบสมการมหลายคาตอบและคาตอบอยในรป )23x

,x(+−

เชน )21

,2(),1,1(),23

,0(

สมการท& 1 32 =+ yx จดสมการ 23

21+−= xy

สมการท& 2 642 =+ yx จดสมการ 23

21+−= xy

ถา 21a a= และ 21b b= ความชนเทากนและจดตดบนแกน Y เปนจดเดยวกน หรอเปนสมการเดยวกน กราฟของเสนตรงทงสองจะทบกนแลว

ดงตวอยางท 5

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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แบบฝกท� 3 การแกระบบสมการเชงเสนโดยใชการแทนคา

1. จงแกระบบสมการเชงเสนตอไปน&โดยใชการแทนคา (ตวอยางท� 1)

1) 7yx4 =+

5yx5 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... 2) 8yx2 =+

16yx3 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... .......................................................................... ...............................................................................

แบบฝกท� 2.2 การแกระบบสมการเชงเสนโดยใชการแทนคา

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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3) 7yx2 =+

13yx3 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... .......................................................................... ............................................................................... .......................................................................... ............................................................................... 4) 2yx7 =+

4yx5 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................. .......................................................................... ............................................................................... .......................................................................... ............................................................................... .......................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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5) 13yx4 =+

6yx3 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... .......................................................................... ............................................................................... .......................................................................... ............................................................................... 6) 5yx2 =+

1yx4 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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2. จงแกระบบสมการเชงเสนสองตวแปรตอไปน&โดยใชการแทนคา (ตวอยางท� 2)

1) 19y5x =+

15y4x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... 2) 3y2x =+

1y3x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... .......................................................................... ............................................................................... ........................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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3) 6y2x =+

14y3x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... .......................................................................... ............................................................................... .......................................................................... ............................................................................... 4) 14y2x =+

19y3x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................. .......................................................................... ............................................................................... .......................................................................... ............................................................................... .......................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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5) 2y3x =+

3y2x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... .......................................................................... ............................................................................... .......................................................................... ............................................................................... 6) 0y3x =+

2y2x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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3. จงแกระบบสมการเชงเสนสองตวแปรตอไปน& โดยใชการแทนคา (ตวอยางท� 3-5)

1) 0yx =+

1yx3 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. 2) 1yx3 =+

9y2x =+− ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. ............................................................................ .............................................................................

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3) 6y2x =+

2y2x =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... 4) 1y3x2 =+

3yx =− ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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5) 2y3x =−

2y6x2 =+− ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... 6) 1x2y +−=

3x2y −−= ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ...............................................................................

ครบญนา เกษ กลมสาระการเรยนรคณตศาสตร

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7) 2yx3 =−

4y2x6 =− ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... 8) 5yx3 =+

10y2x6 =+ ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ...............................................................................

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9) 12y6x8 −=−

2y43

x −=−

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10) 5y32

x =+−

6y6x9 =− ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ........................................................................... ............................................................................... ............................................................................ ............................................................................. .......................................................................... ............................................................................... ............................................................................ .............................................................................