# 4.2 Solve Applications with Systems of Equations

Topics covered in this section are:

## 4.2.1 Solve Direct Translation Applications

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we’ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.

### HOW TO: Solve applications with systems of equations.

1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose variables to represent those quantities.
4. Translate into a system of equations.
5. Solve the system of equations using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

We solved number problems with one variable earlier. Let’s see how differently it works using two variables.

#### Example 1

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Solution

#### Example 2

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $\$25,000$plus$\$15$ for each training session. Option B would pay her $\$10,000+\$40$  for each training session. How many training sessions would make the salary options equal?

Solution

As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.

#### Example 3

Translate to a system of equations and then solve:
When Jenna spent $10$ minutes on the elliptical trainer and then did circuit training for $20$ minutes, her fitness app says she burned $278$ calories. When she spent $20$ minutes on the elliptical trainer and $30$ minutes circuit training she burned $473$ calories. How many calories does she burn for each minute on the elliptical trainer? How many calories for each minute of circuit training?

Solution

## 4.2.2 Solve Geometry Applications

We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.

The measures of two complementary angles add to $90$ degrees. The measures of two supplementary angles add to $180$ degrees.

### COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Two angles are complementary if the sum of the measures of their angles is $90$ degrees.

Two angles are supplementary if the sum of the measures of their angles is $180$ degrees.

If two angles are complementary, we say that one angle is the complement of the other.

If two angles are supplementary, we say that one angle is the supplement of the other.

#### Example 4

Translate to a system of equations and then solve.
The difference of two complementary angles is $26$ degrees. Find the measures of the angles.

Solution

In the next example, we remember that the measures of supplementary angles add to $180$.

#### Example 5

Translate to a system of equations and then solve:
Two angles are supplementary. The measure of the larger angle is twelve degrees less than five times the measure of the smaller angle. Find the measures of both angles.

Solution

Recall that the angles of a triangle add up to $180$ degrees. A right triangle has one angle that is $90$ degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.

#### Example 6

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

Solution

We will draw and label a figure.

Often it is helpful when solving geometry applications to draw a picture to visualize the situation.

#### Example 7

Translate to a system of equations and then solve:
Randall has $125$ feet of fencing to enclose the part of his backyard adjacent to his house. He will only need to fence around three sides, because the fourth side will be the wall of the house. He wants the length of the fenced yard (parallel to the house wall) to be $5$ feet more than four times as long as the width. Find the length and the width.

Solution

## 4.2.3 Solve uniform motion applications

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was $D=rt$ where $D$ is the distance traveled, $r$ is the rate, and $t$ is the time.

Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.

#### Example 8

Translate to a system of equations and then solve:
Joni left St. Louis on the interstate, driving west towards Denver at a speed of $65$ miles per hour. Half an hour later, Kelly left St. Louis on the same route as Joni, driving $78$ miles per hour. How long will it take Kelly to catch up to Joni?

Solution

A diagram is useful in helping us visualize the situation.

Identify and name what we are looking for. A chart will help us organize the data. We know the rates of both Joni and Kelly, and so we enter them in the chart. We are looking for the length of time Kelly, $k$, and Joni, $j$, will each drive.

 Rate $\cdot$ Time $=$ Distance Joni $65$ $j$ $65j$ Kelly $78$ $k$ $78k$

Since $D=r \cdot t$ we can fill in the Distance column.

Translate into a system of equations.

To make the system of equations, we must recognize that Kelly and Joni will drive the same distance. So,

$65j=78k$

Also, since Kelly left later, her time will be $\frac{1}{2}$ hour less than Joni’s time. So,

Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.

Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.

The images below show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water $b$ and the speed of the river current $c$.

The boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving $b+c$.

Now, the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is $b−c$.

We’ll put some numbers to this situation in the next example.

#### Example 9

Translate to a system of equations and then solve.
A river cruise ship sailed $60$ miles downstream for $4$ hours and then took $5$ hours sailing upstream to return to the dock. Find the speed of the ship in still water and the speed of the river current.

Solution

Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in the next example. A wind current in the same direction as the plane is flying is called a tailwind. A wind current blowing against the direction of the plane is called a headwind.

#### Example 10

Translate to a system of equations and then solve:
A private jet can fly $1,095$ miles in three hours with a tailwind but only $987$ miles in three hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Solution