Topics
5. Systems of Equations
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United States of America
Grade 8

5.03 Systems of equations using substitution method

Lesson
Practice

Introduction

We've previously looked at how to  solve systems of equations graphically  . Now, we'll look at how to solve a system of equations algebraically, using the substitution method.

Substitution method

The substitution method relies on the substitution property which states: if a=b then b can be replaced with a in any equation or inequality.

To solve a system of equations by substitution, we can solve one equation for either of the variables, followed by substituting the result into the other equation.

For example, if we want to solve the following system of equations: \begin{cases}x + y &= 7 \\ x - y &= 3 \end{cases}We can solve the first equation for x:

\displaystyle x+y\displaystyle =\displaystyle 7First equation
\displaystyle x+y-y\displaystyle =\displaystyle 7-ySubtract y from both sides
\displaystyle x\displaystyle =\displaystyle 7-ySimplify

We can now substitute this equation into the second equation where we see x:

\displaystyle x-y\displaystyle =\displaystyle 3Second equation
\displaystyle (7-y)-y\displaystyle =\displaystyle 3Replace x with x=7-y from first equation
\displaystyle 7-2y\displaystyle =\displaystyle 3Combine like terms
\displaystyle 7-7-2y\displaystyle =\displaystyle 3-7Subtract 7 from both sides
\displaystyle -2y\displaystyle =\displaystyle -4Simplify
\displaystyle -2y \div -2\displaystyle =\displaystyle -4 \div -2Divide both sides by -2
\displaystyle y\displaystyle =\displaystyle 2Simplify

Now that we've solved for y, we can substitute this value into either equation to solve for x:

\displaystyle x+y\displaystyle =\displaystyle 7First equation
\displaystyle x+2\displaystyle =\displaystyle 7Substitute y=2 into the equation
\displaystyle x+2-2\displaystyle =\displaystyle 7-2Subtract 2 from both sides
\displaystyle x\displaystyle =\displaystyle 5Simplify

We've now been able to solve for x and y which means the solution to our system of equations is \left(5,2\right)

When solving systems of equations with substitution, it can be helpful to do the following:

  • Being organized is key. Name your equations by writing (1) & (2) next to them, and whenever you create new equations out of one or both of them, you can name them (3), \, (4) , etc. When substituting equation (3) into equation (1) you can write it in shorthand as (3) \to (1). This helps to keep your ideas in order and not get confused.
  • When dealing with equations with big numbers, see if you can simplify them before beginning to solve them, eg. 2x - 4 = 6y can be simplified to x - 2 = 3y without changing the values of x & y.

Examples

Example 1

Solve the following system using the substitution method.

  • Equation 1: 3y -x =-4

  • Equation 2: 2 + x - y = 0

Worked Solution
Create a strategy

Solve one linear equation for y in terms of x then substitute that expression for y in the other linear equation.

Apply the idea

The equation 2 looks easier to manipulate. So we solve equation 2 for y in terms of x.

\displaystyle 2 + x - y\displaystyle =\displaystyle 0Write equation 2
\displaystyle 2+ x - y-2\displaystyle =\displaystyle 0-2Subtract 2 from both sides
\displaystyle x-y\displaystyle =\displaystyle -2Evaluate
\displaystyle x-y-x\displaystyle =\displaystyle -2 - xSubtract x from both sides
\displaystyle -y\times -1\displaystyle =\displaystyle (-2-x) \times -1Multiply both sides by -1
\displaystyle y\displaystyle =\displaystyle 2 + xCall this equation 3

Substitute equation 3: y = 2 + x into equation 1: 3y - x = -4

\displaystyle 3y - x\displaystyle =\displaystyle -4Write equation 1
\displaystyle 3(2 + x) - x\displaystyle =\displaystyle -4Substitute the expression for y
\displaystyle 6 + 3x - x\displaystyle =\displaystyle -4Distribute 3 into the parenthesis
\displaystyle 6+3x-x-6\displaystyle =\displaystyle -4-6Subtract 6 from both sides
\displaystyle 2x\displaystyle =\displaystyle -10Simplify
\displaystyle \dfrac{2x}{2}\displaystyle =\displaystyle \dfrac{-10}{2}Divide both sides by 2
\displaystyle x\displaystyle =\displaystyle -5Evaluate

Substitute the value of x into the equation 1.

\displaystyle 3y - x\displaystyle =\displaystyle -4Write the equation 1
\displaystyle 3y - (-5)\displaystyle =\displaystyle -4Substitute the value of x
\displaystyle 3y + 5\displaystyle =\displaystyle -4Simplify
\displaystyle 3y+5-5\displaystyle =\displaystyle -4-5Subtract 5 from both sides
\displaystyle 3y\displaystyle =\displaystyle -9Simplify
\displaystyle \dfrac{3y}{3}\displaystyle =\displaystyle \dfrac{-9}{3}Divide both sides by 3
\displaystyle y\displaystyle =\displaystyle -3Evaluate

The solution of the system is (-5, -3).

Reflect and check

Check your answer. Remember, when you substitute the x and y-values into the equations they must create true statements in order for them to be the solution.

\text{equation}\,1\text{equation}\,2
3y - x = 42 + x - y = 0
3\times(-3) - (-5) = 42 + (-5) - (-3) = 0
-9 + 5 = 4-3 + 3
4 = 40 = 0

Both equations are true. So (-5, -3) is the solution to the system.

Example 2

We want to solve the following system of equations using the substitution method.

  • Equation 1: y = -2x - 1

  • Equation 2: x - 6y = -59

a

First solve for x.

Worked Solution
Create a strategy

Substitute y = -2x - 1 into equation 2.

Apply the idea
\displaystyle x - 6y\displaystyle =\displaystyle -59Write equation 2
\displaystyle x - 6(-2x - 1)\displaystyle =\displaystyle -59Substitute equation 1
\displaystyle x + 12x + 6\displaystyle =\displaystyle -59Distribute -6
\displaystyle 13x+6-6\displaystyle =\displaystyle -59 - 6Subtract 6 from both sides
\displaystyle \dfrac{13x}{13}\displaystyle =\displaystyle \dfrac{-65}{13}Simplify and divide both sides by 13
\displaystyle x\displaystyle =\displaystyle -5Evaluate
b

Solve for y.

Worked Solution
Create a strategy

Substitute the value of x in part (a) into equation 1.

Apply the idea
\displaystyle y\displaystyle =\displaystyle -2x - 1Write equation 1
\displaystyle =\displaystyle -2 ( -5) - 1Substitute the value of x
\displaystyle =\displaystyle 9Evaluate
Idea summary

The substitution method:

  • Solve one linear equation for y in terms of x (or x in terms of y).

  • Substitute that expression for y into the other linear equation.

  • Solve for the value of x.

  • Substitute the value of x into one of the equations to solve for the value of y.

Outcomes

8.EE.C.8

Analyze and solve pairs of simultaneous linear equations.

8.EE.C.8.B

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

8.EE.C.8.C

Solve real-world and mathematical problems leading to two linear equations in two variables.

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