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5. Systems of Equations
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United States of America
Grade 8

5.05 Problem solving with systems of equations

Lesson
Practice

Applications of linear systems

The magic of systems of equations comes to life when we see how useful it is in real life applications. We can use systems when we have at least two unknown quantities and at least two pieces of information involving both of these quantities.

The first step is to use some variables to represent the quantities, followed by figuring out how to write the bits of information down as equations.

We can then choose a method of solving to use, graphically, by substitution, or by elimination based on what's going to be the easiest.

Examples

Example 1

When comparing some test results Christa noticed that the sum of her Geography test score and Science test score was 172, and that their difference was 18.

Given that her Geography score is x and her Science score is y and she scored higher for the Geography test:

a

Use the sum of the test scores to form an equation. We will refer to this as equation 1.

Worked Solution
Create a strategy

Write an equation in terms of x and y.

Apply the idea

The sum of her two test scores is 172, where x and y are the scores.

Equation 1: \quad x + y = 172

b

Use the difference of the test scores to form an equation. We will refer to this as equation 2.

Worked Solution
Create a strategy

Write an equation in terms of x and y.

Apply the idea

The difference between the scores is 18, where x and y are the scores. Since x \gt y we can subtract y from x.

Equation 2:\quad x - y = 18

c

Use these two equations to find her Geography score.

Worked Solution
Create a strategy

We can use the elimination method to solve for x.

Apply the idea

Notice that the coefficient of y in both equations is the same, but with opposite sign. So we can eliminate the variable y by adding the two equations together.

\begin{array}{c} & &x &+ &y &= &172 \\ &+ &x &- &y &= &18 \\ \hline \\ & &2x & & &= &190 \\ & &x & & &= &95 \end{array}

So her Geography score is 95.

d

Now find her Science score.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle x + y\displaystyle =\displaystyle 172Write equation 1
\displaystyle 95 + y\displaystyle =\displaystyle 172Substitute the value of x
\displaystyle 95+y-95\displaystyle =\displaystyle 172-95Subtract 95 from both sides
\displaystyle y\displaystyle =\displaystyle 77Simplify

So her Science score is 77.

Example 2

The perimeter of the triangle below is 56\text{ cm}, and the same values for x and y are used to construct the rectangle shown. The rectangle's length is 8\text{ cm} longer than its width.

A triangle and a rectangle drawn side by side. The triangle has lengths of 4y, 2x and 6. The rectangle has one of the sides measuring x plus 3 and another of side 2y minus 4
a

Write an equation for the perimeter of the triangle. Call this equation 1.

Worked Solution
Create a strategy

The perimeter of the triangle is the sum of the lengths of its three sides.

Apply the idea

The perimeter of the triangle is 56\text{ cm}.

From the figure we have 2x, \, 4y, and 6 units as the length of the sides of the triangle.

So we have Equation 1:

\displaystyle 2x + 4y + 6\displaystyle =\displaystyle 56Write the equation
\displaystyle 2x + 4y\displaystyle =\displaystyle 50Subtract 6 from both sides
b

Use the rectangle to find x in terms of y. Call this equation 2 .

Worked Solution
Create a strategy

Write the expression for the length of the rectangle and the width of the rectangle in terms of x and y and solve for x in terms of y .

Apply the idea

From the figure, 2y - 4 is the length, and x + 3 is the width of the rectangle.

\displaystyle \text{length}\displaystyle =\displaystyle \text{width} + 8Write an equation for the length and width
\displaystyle 2y - 4\displaystyle =\displaystyle (x + 3) + 8Substitute the expressions
\displaystyle x + 11\displaystyle =\displaystyle 2y - 4Simplify
\displaystyle x+11-11\displaystyle =\displaystyle 2y - 4 -11Subtract 11 from both sides
\displaystyle x\displaystyle =\displaystyle 2y - 15Simplify

So we have Equation 2: \quad x = 2y - 15

c

Solve for the value of y.

Worked Solution
Create a strategy

We can use the substition method to solve for y.

Apply the idea
\displaystyle 2x +4y\displaystyle =\displaystyle 50Write the Equation 1
\displaystyle 2(2y-15) + 4y \displaystyle =\displaystyle 50Substitute x from Equation 2 into Equation 1
\displaystyle 4y - 30 + 4y\displaystyle =\displaystyle 50Distribute the parentheses
\displaystyle 8y - 30\displaystyle =\displaystyle 50Simplify
\displaystyle 8y\displaystyle =\displaystyle 50 + 30Add 30 to both sides
\displaystyle \dfrac{8y}{18}\displaystyle =\displaystyle \dfrac{80}{8}Simplify and divide both sides by 8
\displaystyle y\displaystyle =\displaystyle 10Evaluate
d

Finally, use the above value for y to find x.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle x\displaystyle =\displaystyle 2y - 15Write Equation 2
\displaystyle x\displaystyle =\displaystyle 2(10) -15Substitute the value of y
\displaystyle x\displaystyle =\displaystyle 5Evaluate

Example 3

A band plans to record a demo at a local studio.

a

The cost of renting studio A is \$50 plus \$100 per hour.

Write an equation in the form y=mx+b to represent the total cost of hiring the studio, y, as a function of the hours rented, x.

Worked Solution
Create a strategy

The constant represents the down payment on studio A.

The slope represents the cost of renting studio A per hour.

Apply the idea

The down payment on studio A is \$50. The cost of renting the studio per hour is \$100. Substitute these in for b and m respectively.

y=100x+50

b

The cost of renting studio B is \$200 plus \$50 per hour.

Write an equation in the form y=mx+b to represent the total cost of hiring the studio, y, as a function of the hours rented, x.

Worked Solution
Create a strategy

The constant represents the down payment on studio B.

The slope represents the cost of renting studio B per hour.

Apply the idea

The down payment on studio B is \$200. The cost of renting the studio per hour is \$50. Substitute these in for b and m respectively.

y=50x+200

c

Sketch the two lines representing these equations on the graph.

Worked Solution
Create a strategy

To sketch a line we want to find two points on it. We can use the y-intercept and the slope to find a second point. Otherwise, we can find a second point by substituting in a value of x and solving for y.

Apply the idea

Using the y-intercept of the line and substituting the value of x=1 to find the y-coordinate of the second point, we have:

1
2
3
4
5
x
100
200
300
400
500
y
d

What is the solution to the system of equations as an ordered pair (x,y)?

Worked Solution
Create a strategy

There is one point that lies on both lines and therefore satisfies both equations. That point is the point of intersection.

Apply the idea

The point of the intersection is the solution to the system of equations.

The solution is (3,350).

e

What do the coordinates of the solution tell us in context of the problem?

A
The coordinates do not have real life application.
B
Renting studio B could cost \$350 to rent for 3 hours, which is more than it would cost to rent studio A for the same time.
C
Renting either studio A or B for 3 hours costs the same amount, \$350.
D
Renting studio A could cost \$350 to rent for 3 hours, which is more than it would cost to rent studio B for the same time.
Worked Solution
Create a strategy

What does the x-value of the solution represent?

What does the y-value represent?

Apply the idea

The x-value of the solution represents the number of hours for which each studio is used. The y-value represents the cost for using each studio for 3 hours.

The fact that (3, 350) is a solution to both equations means that when x=3, the x-values of both equations are equal to 350.

The answer is option C.

Renting either studio A or B for 3 hours costs the same amount, \$350.

Idea summary

The systems of equations are useful in real-life applications. It is usually used when we have at least two unknown quantities and at least two pieces of information involving both of these quantities.

Once you write your equations to represent the problem, you can decide whether to use the substitution method, elimination method, or a graphical method to solve the equations simultaneously.

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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