Lesson Objectives
• Demonstrate an understanding of how to solve a System of Linear Equations in Two Variables
• Demonstrate an understanding of how to check the solution for a word problem
• Learn the six-step process used to solve a word problem with linear systems

## How to Solve Word Problems with Linear Systems

We previously learned a six-step process used to set up and solve a word problem that involves a linear equation in one variable. We can build on that process and move into a more challenging topic that involves solving word problems that require setting up and solving a system of linear equations. Let's modify our six-step process to work for applications of linear systems.

### Six-step method for Applications of Linear Systems

1. Read the problem, get a clear understanding of the objective
2. Assign a variable to represent each unknown
3. Write two equations using both variables
4. Solve the linear system
5. State the answer using a nice clear sentence
6. Check the result by reading back through the problem
Let's look at a few examples.
Example 1: Solve each word problem.
Molly's school is selling tickets to a dance performance. On the first day of ticket sales, the school sold 9 adult tickets and 12 child tickets for a total of $183. The school took in$160 on the second day by selling 12 adult tickets and 4 child tickets. What is the price each of one adult ticket and one child ticket?
In some cases, it may help to organize the information into a table:
Category Day 1 Day 2
Child Tickets Sold 124
Total Revenue $183$160
Step 1) What is our main objective for this problem? We want to find the cost of one adult ticket along with the cost of one child ticket.
Step 2) Assign a variable for each unknown
let x = cost per adult ticket
then y = cost per child ticket
Step 3) Write two equations using both variables
We have the information for day 1 and day 2 organized in our table above. We can make an equation from each.
Day 1:
9x + 12y = 183
Day 2:
12x + 4y = 160
In our first equation, we are multiplying 9 (number of adult tickets sold) by x ( the cost per adult ticket). We add this to the product of 12 (number of child tickets sold) and y (the cost per child ticket). The result is 183 (total revenue for day 1). The second equation is formed using the same thought process.
Step 4) Solve the linear system
9x + 12y = 183
12x + 4y = 160
Let's label our equations:
1) 9x + 12y = 183
2) 12x + 4y = 160
We will use elimination. Let's multiply equation 2 by (-3). This will give us 12y in equation 1 and (-12y) in equation 2. The y variable will be eliminated when we add our equations together.
2) -36x - 12y = -480
Now we can add the left sides and set this equal to the sum of the right sides: $$\hspace{.25em}+9x + 12y=183$$ $$\underline{-36x - 12y=-480}$$ We can see that y will be eliminated:
12y + (-12y) = 0: $$\require{cancel}\hspace{.5em}+9x + \cancel{12y}=183$$ $$\underline{-36x + \cancel{-12y}=-480}$$ On the left side, we add 9x + (-36x), this gives us -27x. On the right side, we add 183 and (-480), this gives us (-297): $$-27x=-297$$ We can solve this equation for x by dividing each side by -27:
x = 11
Now we can plug in for x in either original equation. Let's plug in for x in equation 1.
9(11) + 12y = 183
99 + 12y = 183
12y = 84
y = 7
Our solution is (11,7).
Step 5) Since our solution is given as (11,7), this means x = 11 and y = 7. Our adult ticket price was represented with x and our child ticket price was represented with y. We can state our answer as:
A child ticket sold for $7, while an adult ticket sold for$11.
Step 6) Check
We can go back through the information in the problem. We end up just checking the equations we made earlier:
9(11) + 12(7) = 183
99 + 84 = 183
183 = 183
12(11) + 4(7) = 160
132 + 28 = 160
160 = 160
Let's look at another problem.
Example 2: Solve each word problem.
A plane traveled 480 miles each way to Roanville and back. The trip there was with the wind and took 3 hours. The trip back was into the wind and took 4 hours. What is the speed of the plane in still air? What is the speed of the wind?
In some cases, it may help to organize the information into a table:
Category Trip There Trip Back
Distance (miles) 480480
Rate (mph) ??
Time (hours) 34
Step 1) What is our main objective for this problem? We want to find the speed of the plane in still air, along with the wind speed.
Step 2) Assign a variable for each unknown
let x = speed of the plane in still air
then y = wind speed
Let's update our table using the assigned variables:
Category Trip There Trip Back
Distance (miles) 480480
Rate (mph) x + yx - y
Time (hours) 34
When the wind is with us, we add the wind speed to the speed of the plane in still air. This gives us (x + y) when going with the wind. When the wind is going against us, we subtract the wind speed from the speed of the plane in still air. This gives us (x - y) when going against the wind.
Step 3) Write two equations using both variables
The distance formula states that d = r x t. We know our distance will be equal to our rate of speed multiplied by the amount of time traveled:
3(x + y) = 480
4(x - y) = 480
Step 4) Solve the linear system
Let's label our equations:
1) 3(x + y) = 480
2) 4(x - y) = 480
We will first simplify each equation:
1) 3x + 3y = 480
2) 4x - 4y = 480
We will probably have an easier time using substitution in this case. Let's solve equation 1 for y:
1) 3x + 3y = 480
3y = -3x + 480
y = -x + 160
Now we can plug in (-x + 160) for y in equation 2:
2) 4x - 4y = 480
4x - 4(-x + 160) = 480
4x + 4x - 640 = 480
8x = 1120
x = 140
Now we can plug in 140 for x in either original equation. Let's plug into equation 1:
3(140) + 3y = 480
420 + 3y = 480
3y = 60
y = 20
Our solution is the ordered pair (140,20).
Step 5) Since our solution is given as (140,20), this means x = 140 and y = 20. The speed of the plane in still air was represented with x and the wind speed was represented with y. We can state our answer as:
The plane's speed in still air is 140 mph, while the speed of the wind is 20 mph.
Step 6) Check
We can go back through the information in the problem. We end up just checking the equations we made earlier:
3(140 + 20) = 480
3(160) = 480
480 = 480
4(140 - 20) = 480
4(120) = 480
480 = 480

#### Skills Check:

Example #1

Solve each word problem.

Jason and Mallorie are selling baked goods for a school fundraiser. Students can buy boxes of assorted donuts and boxes of cupcakes. Jason sold 3 boxes of donuts and 14 boxes of cupcakes for a total of $178. Mallorie sold 14 boxes of donuts and 6 boxes of cupcakes for a total of$178. What is the cost each of one box of assorted donuts and one box of cupcakes?

A
Donuts: $14, Cupcakes$11
B
Donuts: $5, Cupcakes$9
C
Donuts: $8, Cupcakes$18
D
Donuts: $8, Cupcakes$11
E
Donuts: $9, Cupcakes$15

Example #2

Solve each word problem.

A boat traveled 119 miles each way downstream and back. The trip downstream took 7 hours. The trip back took 17 hours. What is the speed of the boat in still water? What is the speed of the current?

A
Boat 3mph, Current 1mph
B
Boat 2mph, Current 6mph
C
Boat 11mph, Current 6mph
D
Boat 6mph, Current 6mph
E
Boat 12mph, Current 5mph

Example #3

Solve each word problem.

The senior class at Malcolm High School and the junior class at Ridge Point High School both planned trips to the city park. The senior class at Malcolm High School rented and filled 13 vans and 10 buses with 369 students. Ridge Point High School rented and filled 18 vans and 16 buses with 554 students. Each van carried the same number of students; additionally, each bus carried the same number of students. Find the number of students in each van and in each bus.

A
Van 6, Bus 27
B
Van 9, Bus 28
C
Van 10, Bus 12
D
Van 13, Bus 20
E
Van 11, Bus 15