5.2 Solving Systems of Equations by Substitution

Learning objectives

Learning objectives

• Solve a system of equations by substitution

• Solve applications of systems of equations by substitution

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

In this section, we will solve systems of linear equations by the substitution method.

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in Example 5.13.

Example 5.13

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in Example 5.14.

Example 5.14

Solution

The second equation is already solved for y. We will substitute the expression in place of y in the first equation.

If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y.

Example 5.15

Solution

We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

In Example 5.15 it was easiest to solve for y in the first equation because it had a coefficient of 1. In Example 5.16 it will be easier to solve for x.

Example 5.16

Solution

We will solve the first equation for $x$ and then substitute the expression into the second equation.

When both equations are already solved for the same variable, it is easy to substitute!

Example 5.17

Solution

Since both equations are solved for y, we can substitute one into the other.

Be very careful with the signs in the next example.

Example 5.18

Solution

We need to solve one equation for one variable. We will solve the first equation for y.

In Example 5.19, it will take a little more work to solve one equation for x or y.

Example 5.19

Solution

We need to solve one equation for one variable. We will solve the first equation for x.

Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.

Look back at the equations in Example 5.19. Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

Example 5.20

Solution

The second equation is already solved for y, so we can substitute for y in the first equation.

Since 0 = −10 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

Example 5.21

Solution

In the Example 5.22, we’ll use the formula for the perimeter of a rectangle, P = 2*L* + 2*W*.

Example 5.22

Solution

For Example 5.23 we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

Example 5.23

Solution

We will draw and label a figure.

Example 5.24

Solution

Section 5.2 Exercises

Section 5.2 Exercises

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

Solve Applications of Systems of Equations by Substitution

In the following exercises, translate to a system of equations and solve.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} -2x+y=-11 \\ x+3y=9 \\ \end{array}$

2. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} x+3y=10 \\ 4x+y=18 \\ \end{array}$

3. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} x+y=6 \\ y=3x-2 \\ \end{array}$

4. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 2x-y=1 \\ y=-3x-6 \\ \end{array}$

1. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 4x+y=2 \\ 3x+2y=-1 \\ \end{array}$

2. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} -x+y=4 \\ 4x-y=2 \\ \end{array}$

3. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} x-5y=13 \\ 4x-3y=1 \\ \end{array}$

4. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} x-6y=-6 \\ 2x-4y=4 \\ \end{array}$

1. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} y=3x-16 \\ y=\frac{1}{3}x \\ \end{array}$

2. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} y=-x+10 \\ y=\frac{1}{4}x \\ \end{array}$

3. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} x-4y=-4 \\ -3x+4y=0 \\ \end{array}$

4. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 4x-y=0 \\ 2x-3y=5 \\ \end{array}$

1. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 2x-3y=12 \\ -12y+8x=48 \\ \end{array}$

2. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 5x+2y=12 \\ -4y-10x=-24 \\ \end{array}$

3. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 3x+2y=9 \\ y=-\frac{3}{2}x+1 \\ \end{array}$

4. Solve the system by substitution. $\mathrm{\{}\begin{array}{l} 5x-3y=2 \\ y=\frac{5}{3}x-4 \\ \end{array}$

1. The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

2. The sum of two number is −6. One number is 10 less than the other. Find the numbers.

3. The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

4. The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

1. The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

2. The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

3. Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of$100 for each policy sold. The second pays a salary of $20,000 plus a commission of$50 for each policy sold. How many policies would need to be sold to make the total pay the same?

4. Kenneth currently sells suits for company A at a salary of $22,000 plus a$10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a$4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

1. $\mathrm{\{}\begin{array}{l} 2x+y=-4 \\ 3x-2y=-6 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} x-2y=-5 \\ 2x-3y=-4 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 5x-2y=-6 \\ y=3x+3 \\ \end{array}$

4. $\mathrm{\{}\begin{array}{l} 2x+3y=3 \\ y=-x+3 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 2x+5y=1 \\ y=\frac{1}{3}x-2 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 3x-2y=6 \\ y=\frac{2}{3}x+2 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 2x+y=10 \\ -x+y=-5 \\ \end{array}$

4. $\mathrm{\{}\begin{array}{l} 3x+y=1 \\ -4x+y=15 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} x+3y=1 \\ 3x+5y=-5 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 2x+y=5 \\ x-2y=-15 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y=-2x-1 \\ y=-\frac{1}{3}x+4 \\ \end{array}$

4. $\mathrm{\{}\begin{array}{l} y=2x-8 \\ y=\frac{3}{5}x+6 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 4x+2y=8 \\ 8x-y=1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 15x+2y=6 \\ -5x+2y=-4 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y=3x \\ 6x-2y=0 \\ \end{array}$

4. $\mathrm{\{}\begin{array}{l} 2x+16y=8 \\ -x-8y=-4 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y=-4x \\ 4x+y=1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} y=\frac{7}{8}x+4 \\ -7x+8y=6 \\ \end{array}$

3. The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

4. The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

1. The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

2. The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

3. The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

4. The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

1. Jackie has been offered positions by two cable companies. The first company pays a salary of $14,000 plus a commission of$100 for each cable package sold. The second pays a salary of $20,000 plus a commission of$25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

2. Mitchell currently sells stoves for company A at a salary of $12,000 plus a$150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a$50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

3. Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system $\mathrm{\{}\begin{array}{l} 56s=70t \\ s=t+\frac{1}{2} \\ \end{array}$.

1. ⓐ for $t$ to find out how long it will take Tina to catch up to Stephanie.

2. ⓑ what is the value of $s$, the number of hours Stephanie will have driven before Tina catches up to her?

4. Solve the system of equations$\mathrm{\{}\begin{array}{l} 3x+y=12 \\ x=y-8 \\ \end{array}$ by substitution and explain all your steps in words.