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 which 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.

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:

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:

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:

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

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

- Read the problem, get a clear understanding of the objective
- Assign a variable to represent each unknown
- Write two equations using both variables
- Solve the linear system
- State the answer using a nice clear sentence
- Check the result by reading back through the problem

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 |
---|---|---|

Adult Tickets Sold | 9 | 12 |

Child Tickets Sold | 12 | 4 |

Total Revenue | $183 | $160 |

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) | 480 | 480 |

Rate (mph) | ? | ? |

Time (hours) | 3 | 4 |

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) | 480 | 480 |

Rate (mph) | x + y | x - y |

Time (hours) | 3 | 4 |

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

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