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3. Systems of Linear Equations & Inequalities
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Integrated Math 1

3.03 Solving systems of equations by elimination

Lesson
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Lesson

Concept summary

If at least one pair of variables in a system of equations can be quickly made to have the same coefficient, or already have the same coefficient, the system can be solved efficiently by using elimination.

Elimination method

A method of solving a system of equations by adding or subtracting the equations until only one variable remains

Worked examples

Example 1

Solve the following system of equations by using the elimination method: \begin{cases} 9x+y=62\\5x+y=38 \end{cases}

Approach

We first want to number our equations to make it easier to work with.

1\displaystyle 9x + y\displaystyle =\displaystyle 62
2\displaystyle 5x + y\displaystyle =\displaystyle 38

Since the coefficents of y are the same with the same sign, we can start by subtracting 2 from 1 to eliminate y from the equation. We can then solve for x, and substitute this value back into one of the equations to solve for y.

Solution

\displaystyle \left(9x + y\right) - \left(5x + y\right)\displaystyle =\displaystyle 62 - 38Subtract equation 2 from equation 1
\displaystyle 4x\displaystyle =\displaystyle 24Combine like terms
\displaystyle x\displaystyle =\displaystyle 6Divide both sides by 4
\displaystyle 5\left(6\right) + y\displaystyle =\displaystyle 38Substitute back into equation 2
\displaystyle 30 + y\displaystyle =\displaystyle 38Evaluate the multiplication
\displaystyle y\displaystyle =\displaystyle 8Subtract 30 from both sides

So the solution to the system of equations is x = 6, y = 8.

Reflection

We could also have solved this by subtracting 1 from 2. We could also have substituted x=6 back into equation 1.

\displaystyle \left(5x + y\right) - \left(9x + y\right)\displaystyle =\displaystyle 38 - 62Subtract equation 1 from equation 2
\displaystyle -4x\displaystyle =\displaystyle -24Combine like terms
\displaystyle x\displaystyle =\displaystyle 6Divide both sides by -4
\displaystyle 9\left(6\right) + y\displaystyle =\displaystyle 62Substitute back into equation 1
\displaystyle 54 + y\displaystyle =\displaystyle 62Evaluate the multiplication
\displaystyle y\displaystyle =\displaystyle 8Subtract 54 from both sides

It doesn't matter which way we choose, but one method will usually be preferable. In this case we didn't have to deal with negative numbers when using the first approach.

Example 2

When comparing test results, Verna noticed that the sum of her Chemistry and English test scores was 128, and that their difference was 16. She scored higher on her Chemistry test.

a

Write a system of equations for this scenario, where x represents Verna's Chemistry test score and y represents her English test score.

Approach

Since we know that Verna's Chemistry test score is higher than her English test score, we should express the difference of the two scores as x - y to get a positive result.

Solution

1\displaystyle x + y\displaystyle =\displaystyle 128
2\displaystyle x - y\displaystyle =\displaystyle 16
b

Solve the system of equations to find her test scores.

Approach

Since the equations have the same coefficient of y with opposite sign, we can add the two equations to eliminate y and solve for x first.

Solution

\displaystyle \left(x + y\right) + \left(x - y\right)\displaystyle =\displaystyle 128 + 16Add equation 1 and equation 2 together
\displaystyle 2x\displaystyle =\displaystyle 144Combine like terms
\displaystyle x\displaystyle =\displaystyle 72Divide both sides by 2
\displaystyle 72 + y\displaystyle =\displaystyle 128Substitute back into equation 1
\displaystyle y\displaystyle =\displaystyle 56Subtract 72 from both sides

So Verna scored 72 on her Chemistry test and 56 on her English test.

Reflection

In this case, the two equations also had the same coefficient of x with the same sign. So we could have instead approached this by subtracting the two equations and solving for y first.

c

Does the solution make sense in terms of the context? Explain your answer.

Solution

Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain.

Outcomes

M1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

M1.A.REI.C.3

Write and solve a system of linear equations in real-world context.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

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