Lesson 22 - Solving Linear Equations By Elimination Via ... · 8/8/2019  · The equations will have two like terms that are opposites. Step 1 here is done. Write the solution as

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Step 2 Step 3 Step 4

How can you tell if a linear system has a variable that can be eliminated by adding? The equations will have two like terms that are opposites. Step 1 here is done.

Write the solution

as an ordered pair.

(4, 2)

Lesson 22 - Solving Linear Equations By Elimination Via Addition OnlySolving Systems of Equations by Elimination

Step 1 Ensure that both equations have a term (with a variable) that are opposites to each other.

Step 2 Add the equations to eliminate one variable, and then solve for the other variable.

Step 3 Substitute the value into either original equation to find the value of the eliminated variable.

Step 4 Write the values from Steps 1 and 2 in an ordered pair (x, y).

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−2𝑥 + 𝑦 = 8−2𝑥 + (𝟐) = 8

− 2 − 2−2𝑥 = 6

𝒙 =

Step 3

Step 4

First, multiply the whole line by –1 to flip all the signs

They become:

2𝑥 + 6𝑦 = 6−2𝑥 + 𝑦 = 8

Add the equations.

2𝑥 + 6𝑦 = 6−2𝑥 + 𝑦 = 8

𝒚

Step 2

Step 1

Solving Systems of Equations by EliminationStep 1 Ensure that both equations have a term (with a variable) that are opposites to each other.

Step 2 Add the equations to eliminate one variable, and then solve for the other variable.

Step 3 Substitute the value into either original equation to find the value of the eliminated variable.

Step 4 Write the values from Steps 1 and 2 in an ordered pair (x, y).

Step 3

Step 2

Step 4

Step 1(if needed)

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Step 1Step 2

Step 3

Perimeter

Difference between twice the length and twice the width

Real-World Applications

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Total number of rentals, between both video games & movies

Total value of all the rentals

$2 per video game

$1 per movie

𝒗 +𝒎 = 𝟏𝟏𝟒𝟐𝒗 +𝒎 = 𝟏𝟕𝟕

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Jennifer and Wilbur each improved their yards by planting hostas and geraniums. They bought their supplies from

the same store. Jennifer spent $128 on 13 hostas and 1 geranium. Wilbur spent $92 on 9 hostas and 1 geranium.

What is the cost of one hosta and the cost of one geranium? (Hint: Let h be the cost of one hosta, and g be the

cost of 1 geranium.)

Equation #1

Equation #2

Equation #1

Equation #2

What do you do if all 4 variables have coefficients, and none can be eliminated? For example:

𝟑𝒙 + 𝟖𝒚 = 𝟕𝟐𝒙 − 𝟐𝒚 = −𝟏𝟎

The first term of both equations, 3x and 2x, can’t be added together, even if you multiply one of them by –1.The second term of both equations, 8y and –2y, can’t be added together.

But – we can multiply ONE ENTIRE equation by a number, which will make one of the coefficients become an opposite.

Specifically: If we multiply the whole 2nd equation by 4, here’s what it becomes:

𝟒 𝟐𝒙 − 𝟐𝒚 = −𝟏𝟎 ⇒ 𝟖𝒙 − 𝟖𝒚 = −𝟒𝟎

Did you notice that the new 2nd term, −𝟖𝒚, is the opposite of the 2nd term of the 1st equation, 𝟖𝒚 ?Here’s how they line up:

𝟑𝒙 + 𝟖𝒚 = 𝟕𝟖𝒙 − 𝟖𝒚 = −𝟒𝟎Now you can add the 2nd terms!

Solving Linear Equations By Elimination Via Multiplication and Addition

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P. 519Solving Systems of Equations by Multiplying FirstStep 1 Decide which variable to eliminate.Step 2 Multiply one or both equations by a constant so that adding the equations will eliminate the variable.Step 3 Solve the system using the elimination method.

What do you do if all 4 variables have coefficients, and multiplying one whole equation doesn’t help?For example:

−𝟑𝒙 + 𝟐𝒚 = 𝟒𝟒𝒙 − 𝟏𝟑𝒚 = 𝟓

Multiplying the 1st term of the 1st equation by anything won’t give you –4.Multiplying the 1st term of the 2nd equation by anything won’t give you 3.Multiplying the 2nd term of the 1st equation by anything won’t give you 13.Multiplying the 2nd term of the 2nd equation by anything won’t give you –2.

But – we can multiply BOTH ENTIRE equations – each by a different number, which will make two of the coefficients become opposites.

For example: If we multiply the whole 1st equation by 4, here’s what it becomes:

𝟒 −𝟑𝒙 + 𝟐𝒚 = 𝟒 ⇒ −𝟏𝟐𝒙 + 𝟖𝒚 = 𝟏𝟔

And if we multiply the whole 2nd equation by 3, here’s what it becomes:

𝟑 𝟒𝒙 − 𝟏𝟑𝒚 = 𝟓 ⇒ 𝟏𝟐𝒙 − 𝟑𝟗𝒚 = 𝟏𝟓

Did you notice that both new 1st terms are opposites?Here’s how they line up:

−𝟏𝟐𝒙 + 𝟖𝒚 = 𝟏𝟔𝟏𝟐𝒙 − 𝟑𝟗𝒚 = 𝟏𝟓

Now you can add the 1st terms!

P. 519

P. 519Solving Systems of Equations by Multiplying FirstStep 1 Decide which variable to eliminate.Step 2 Multiply one or both equations by a constant so that adding the equations will eliminate the variable.Step 3 Solve the system using the elimination method.

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Equation #1

Equation #2

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A jar containing only nickels and dimes contains a total of 14 coins. The value of all the coins in the jar

is $1.10. How many nickels and dimes are in the jar?