Algebra-2 Section 1-3 And Section 1-4. Quiz 1-2 1. Simplify 1. Simplify -4y – x + 10x + y 2. Is x = -2 a solution to following equation? 3. Solve.

Algebra-2 Algebra-2

Section 1-3Section 1-3

AndAnd

Section 1-4Section 1-4

Quiz 1-2Quiz 1-21. Simplify 1. Simplify -4y – x + 10x + y

2. Is x = -2 a solution to following equation?2. Is x = -2 a solution to following equation?

4143 x3. Solve.3. Solve.

423 x

1-3: Solve Linear 1-3: Solve Linear EquationsEquations

VocabularyVocabularyLinear EquationLinear Equation in in oneone variable. variable.

44xx –– 22 == 66

Definitions:Definitions:Solution to an equationSolution to an equation: the value of the variable : the value of the variable

that makes the equation “true”.that makes the equation “true”.

Equivalent equationEquivalent equation: has the same solution as : has the same solution as the original equation:the original equation:

1024 x 84 xThe solution to both equations is x = 2.The solution to both equations is x = 2.

They are equivalent equations.They are equivalent equations.

Equality Properties Equality Properties

AdditionAddition Property of Equality Property of Equality

SubtractionSubtraction Property of Equality Property of Equality

Multiplication Multiplication Property of EqualityProperty of Equality

DivisionDivision Property of Equality Property of Equality

Only apply to Only apply to equationsequations!!!!!!

““+, -, x, ÷” by the same number on +, -, x, ÷” by the same number on both sidesboth sides of of the the equal signequal sign and you are guaranteed that the and you are guaranteed that the next equation is an next equation is an equivalent equation.equivalent equation.

VocabularyVocabularyCoefficientCoefficient The number that multiplies a variable The number that multiplies a variable

or is “in front of” the variableor is “in front of” the variable

x3

2132 x

2x = 242x = 24

Property ofProperty of Equality Equality

Equivalent equationEquivalent equation to the one above. to the one above.

1x = 121x = 12

÷ 2÷ 2 ÷ 2÷ 2

+3+3 +3+3

Property ofProperty of Equality Equality and and

Turn Turn coefficientscoefficients into into onesones and and addendsaddends into into zeroeszeroes so that they disappear!so that they disappear!

Equivalent equationEquivalent equation to the both to the both equations above.equations above.

inverse prop. Of multiplicationinverse prop. Of multiplication

1. 1. 2 = 3 + x2 = 3 + x

Your turnYour turn: : solve the equation, justify each stepsolve the equation, justify each step

Your turnYour turn: : solve the equation. Justify each step.solve the equation. Justify each step.

2. 2. 12 = 3x + x12 = 3x + x

Your turnYour turn: : solve the equation. Justify each step.solve the equation. Justify each step.

3. 3. -27 = 2x – 3 + 2x-27 = 2x – 3 + 2x

Your turn:Your turn:

6. 6.

55. - 4 = -85

2x183 x

4. 4. = -2= -23

x

Variable on Both SidesVariable on Both Sides24 - x = 3x

+ x+ x + x + x

÷ 4

66

÷ 4

= x= x

Eliminate variable from one side.Eliminate variable from one side.

24 = 4x

The solution looks like: The solution looks like: x = numberx = number

7. 7. 12x = 3 + 3x

Your turn: Solve the following equationsYour turn: Solve the following equations

8.8. 5225 xx

Solving Equations using the Solving Equations using the Distributive Distributive PropertyProperty

3(5x – 8) = -2(-x +7) – 12x3(5x – 8) = -2(-x +7) – 12x

15x -2415x -24 == 2x - 142x - 14 - 12x- 12x

Eliminate parentheses using distributive property.Eliminate parentheses using distributive property.

Combine “like terms”.Combine “like terms”.

15x -2415x -24 == -10x - 14-10x - 14

Solving Equations using the Solving Equations using the Distributive Distributive PropertyProperty

15x -2415x -24 == -10x - 14-10x - 14

Solve using properties of equality. Solve using properties of equality.

+10x+10x

== -14-14

+10x+10x

25x -2425x -24

==

+24+24

25x25x+24+24

1010÷25÷25 ÷25÷25 5

2

25

10x

Another exampleAnother example

)1(4)2(3 xx

)1(463 xx

4463 xx+ 4x+ 4x+ 4x + 4x

+67x7x

÷ 7

= 10= 10

7x – 6 = 4+6

7x7x = 10= 10

÷ 7

7

10 x

Your Turn: Solve using the Your Turn: Solve using the Distributive PropertyDistributive Property

9.9. )12(2)3(2 xx

10.10. )72()2(5 xx

11. 11. 1)23(3)3( xx

12. 12.

Your turn: Solve the following equationsYour turn: Solve the following equations

13.13.

14.14.

)21(3432 xx

xxxx 3)32(5)32(3

)5(5)42( xx

(Get rid of parentheses 1(Get rid of parentheses 1stst using the distributive property.) using the distributive property.)

Solving a multi-variable Solving a multi-variable equationequation 12243 zyx

There are an There are an infinite number of infinite number of combinations.combinations.

x = 0 x = 0 y = 2y = 2 z = 2z = 2

x = 2 x = 2 y = 1y = 1 z = 2z = 2

x = 4 x = 4 y = 0y = 0 z = 0z = 0

12)2(2)2(4)0(3

12)0(2)0(4)4(3

12)1(2)1(4)2(3

Solving a multi-variable Solving a multi-variable equationequation 12243 zyx

x = ? x = ? y = 1y = 1 z = 2z = 2

Could you find the value of ‘x’ if I gave you the valuesCould you find the value of ‘x’ if I gave you the values of ‘y’ and ‘z’ ?of ‘y’ and ‘z’ ? 12243 zyx

12) (2) (43 x122) (21) (43 x

12443 x1283 x

43 x4

3x

Your Turn: Solve for xYour Turn: Solve for x

15. 15. y = 2, z = 1, x = ?y = 2, z = 1, x = ? zyx 32

16. 16. x = 1, y = 3, z = ?x = 1, y = 3, z = ? 6323 zyx

ReviewReviewWhat are the What are the +/-/x/÷ properties of equality+/-/x/÷ properties of equality ? ?

What do the What do the +/-/x/÷ properties of equality+/-/x/÷ properties of equality guarrantee ? guarrantee ?

““add 2 to add 2 to equationequation on on both sidesboth sides of the equal sign”, etc. of the equal sign”, etc.

The resulting equation will be The resulting equation will be equivalentequivalent to the original to the original equation (it will have the same solution).equation (it will have the same solution).

Section 1-4 Section 1-4

Rewrite Formulas and Rewrite Formulas and Equations.Equations.

VocabularyVocabulary

Solve for a variable (more then one variable in the equation)Solve for a variable (more then one variable in the equation): : Use properties of equality to rewrite the equation as an Use properties of equality to rewrite the equation as an equivalentequivalent equation with the equation with the specifiedspecified variable on one side of variable on one side of the equal sign and all other terms on the other side. the equal sign and all other terms on the other side.

Solve the single variable equationSolve the single variable equation: : Use properties of Use properties of equality to rewrite the equation as an equality to rewrite the equation as an equivalentequivalent equation equation with the variable on one side of the equal sign and a with the variable on one side of the equal sign and a number on the other side. number on the other side.

x + 1 = 5x + 1 = 5

= 4= 4x =x =

- 1- 1 - 1- 1

Solve for “x”Solve for “x”

Solve for the variableSolve for the variable: : Use properties of Use properties of equality to rewrite the equation as an equality to rewrite the equation as an equivalentequivalent equation with the variable on equation with the variable on one side of the equal sign and a number one side of the equal sign and a number on the other side. on the other side.

Solve for ‘x’Solve for ‘x’

4 + 2x + y = 64 + 2x + y = 6

÷ 2 ÷ 2÷ 2 ÷ 2

2

2 yx

- 4 - 4- 4 - 4

2x + y = 22x + y = 2

- y - y- y - y

2x = 2 – y 2x = 2 – y

Solve for the variableSolve for the variable: : Use properties of equality to rewrite Use properties of equality to rewrite the equation as an the equation as an equivalentequivalent equation with the equation with the specifiedspecified variable on one side of the equal sign and all other terms variable on one side of the equal sign and all other terms on the other side. on the other side.

Solve for “x”Solve for “x”yx – 2 = 4yx – 2 = 4

+2 +2+2 +2

yx

6

Solve for the variableSolve for the variable: : Use properties of equality to rewrite Use properties of equality to rewrite the equation as an the equation as an equivalentequivalent equation with the equation with the specifiedspecified variable on one side of the equal sign and all other terms variable on one side of the equal sign and all other terms on the other side. on the other side.

÷ y ÷ y÷ y ÷ y yx = 6yx = 6

Your turn:Your turn:

17. 17. Solve for ‘k’Solve for ‘k’

18. 18. Solve for ‘k’Solve for ‘k’

19. 19. Solve for ‘k’Solve for ‘k’

xyk

42

37

734 kym

532 mk

VocabularyVocabulary

FormulaFormula: An equation that relates two or more : An equation that relates two or more quantitiesquantities, , usually usually represented byrepresented by variablesvariables..

wlArea *rectangle

QuantityQuantity: An measure of a real world physical property : An measure of a real world physical property (length, width, temperature, pressure, weight, mass, etc.). (length, width, temperature, pressure, weight, mass, etc.).

W W

L

LwLP 22rectangle

Your turnYour turn: for the area of a : for the area of a triangle formulatriangle formula: :

(‘A’ is a function of ‘b’ and ‘h’.)(‘A’ is a function of ‘b’ and ‘h’.)

bhA2

1

20. 20. Solve for “b”Solve for “b”

21. 21. Solve for “h”. Solve for “h”.

We call this new version of the formulaWe call this new version of the formula “ “b” is a function of “h” and “A”b” is a function of “h” and “A”

22. 22. What do you call this new version of the What do you call this new version of the formula? (similar to: ‘A’ is a function of ‘b’ and formula? (similar to: ‘A’ is a function of ‘b’ and ‘h’.)‘h’.)

Your turn:Your turn:

wlArea *rectangle

W W

L

L

wLP 22rectangle

23. 23. The width of a rectangle is 2 feet. The length is twice The width of a rectangle is 2 feet. The length is twice the width. What is the the width. What is the perimeterperimeter of the rectangle? of the rectangle?

24. 24. The width of a rectangle is 3 feet. The length is four The width of a rectangle is 3 feet. The length is four times the width. What is the times the width. What is the areaarea of the rectangle? of the rectangle?

Formulas are used Formulas are used extensivelyextensively in in science.science.Science and math come together when mathematical Science and math come together when mathematical equations are used to describe the physical world. equations are used to describe the physical world.

Once a formula is known then scientists can use the Once a formula is known then scientists can use the equation to predict the value of unknown variables in the equation to predict the value of unknown variables in the formula. formula.

Solve for radiusSolve for radius

We will now solve for “r”We will now solve for “r”

In this form, we say thatIn this form, we say that ‘ ‘cc’ is a function of ‘’ is a function of ‘rr’.’.rC 2

2

Cr

÷ 2÷ 2ππ ÷ 2 ÷ 2ππ

rC

2

In this form, we say thatIn this form, we say that ‘ ‘rr’ is a function of ‘’ is a function of ‘cc’.’.

Your Turn:Your Turn: hbbA 212

1

25. 25. Solve for ‘h’.Solve for ‘h’.

26. 26. Solve forSolve for

(Area of a trapezoid(Area of a trapezoid: : where the lengthwhere the length of the parallel bases areof the parallel bases are and the distance between them is ‘h’.) and the distance between them is ‘h’.)

21 .... bandb

2b

2b

1b

h

What if two terms have the variable What if two terms have the variable you’re trying to solve for?you’re trying to solve for?Solve the equation for “y”Solve the equation for “y”..

Use “reverse distributive propertyUse “reverse distributive property

9y + 6xy = 309y + 6xy = 30

3069 xyy

÷(9 + 6x) ÷ (9 + 6x)÷(9 + 6x) ÷ (9 + 6x)

)69(

30

xy

30)69( xy

What is “common” to both of What is “common” to both of the left side terms?the left side terms?

““Factor out” the common termFactor out” the common term

““same thing left/right”same thing left/right”

ExampleExample

1023 xxySolve for ‘x’.Solve for ‘x’.

‘‘x’ is common to both terms x’ is common to both terms factor it out (reverse distributive factor it out (reverse distributive

property).property).

10)23( yx

How do you turn (3y – 2) intoHow do you turn (3y – 2) into a “one” so that it disappearsa “one” so that it disappears on the left side of the equation?on the left side of the equation?

÷(3y – 2) ÷(3y – 2)÷(3y – 2) ÷(3y – 2)

)23(

10

yx

403 xxy27. 27. Solve for ‘x’.Solve for ‘x’.

Your turn: Your turn:

28. 28. Solve for ‘y’.Solve for ‘y’. 524 xxy

Solving formula ProblemsSolving formula ProblemsThe perimeter of a rectangular back yard is 41 feet. Its The perimeter of a rectangular back yard is 41 feet. Its length is 12 feet. What is its width?length is 12 feet. What is its width?

?rectangle Perimeter Wft 41P

ft 12wLP 22rectangle

Draw the pictureDraw the pictureWrite the formulaWrite the formula

Replace known variables in Replace known variables in the formula with constantsthe formula with constants

w2ft) 12(2ft 41

w2 24 41 -24 -24-24 -24

÷2 ÷2÷2 ÷2

w2 71

w2 71

w ft 2

71 ft 5.8 ft

2

71w

Solve for the Solve for the variablevariable

Solving formula ProblemsSolving formula Problems1. Draw the picture (it helps to see it)1. Draw the picture (it helps to see it)

2. Write the formula2. Write the formula

3. Replace known variables in the formula with constants3. Replace known variables in the formula with constants

4. Solve for the variable4. Solve for the variable

29. 29. If the base of a triangle is If the base of a triangle is 4 inches 4 inches and its area and its area is is 15 square inches15 square inches, what is its height?, what is its height?

31. 31. The perimeter of a rectangle is The perimeter of a rectangle is 100 miles100 miles. It is . It is 22 miles 22 miles long. How wide is the rectangle?long. How wide is the rectangle?

Your turn: Your turn:

30. 30. The area of a trapezoid is The area of a trapezoid is 40 square feet40 square feet. The length . The length of one base is of one base is 8 feet 8 feet and its height is and its height is 3 feet3 feet, what is the , what is the length of the other base? length of the other base?

wLP 22rectangle bhAreatriangle 21 hbbA 21trapezoid 2

1