4.2 Solve Applications with Systems of Equations

Learning objectives

Learning objectives

• Solve direct translation applications

• Solve geometry applications

• Solve uniform motion applications

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.

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

Example 4.14

Solution

Example 4.15

Solution

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

Example 4.16

Solution

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.

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 4.17

Solution

Step 1. Read the problem.

Step 2. Identify what we are looking for. · We are looking for the measure of each angle.

Step 3. Name what we are looking for. · $\begin{array}{l} \mathrm{Let}\,x=\,\mathrm{the measure of the first angle.} \\ y=\,\mathrm{the measure of the second angle} \\ \end{array}$ Step 4. Translate into a system of equations. · The angles are complementary.

$\,x+y=90$

The difference of the two angles is 26 degrees.

$\,x-y=26$

The system is shown. · $\,\mathrm{\{}\begin{array}{l} x+y=90 \\ x-y=26 \\ \end{array}$ Step 5. Solve the system of equations by elimination. · $\,\begin{array}{l} \underset{\mathrm{\_\_\_\_\_\_\_\_\_\_\_}}{\mathrm{\{}\begin{array}{l} x+y=90 \\ x-y=26 \\ \end{array}} \\ 2x\,=116 \\ \end{array}$ Substitute $x=58$ into the first equation. · $\,\begin{array}{r} x=58 \\ x+y=90 \\ 58+y=90 \\ y=32 \\ \end{array}$ Step 6. Check the answer in the problem.

$\,\begin{array}{l} 58+32=90✓ \\ 58-32=26✓ \\ \end{array}$

Step 7. Answer the question. · The angle measures are 58 and 32 degrees.

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

Example 4.18

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 4.19

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 4.20

Solution

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 4.21

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.

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,

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

$\,k=j-\frac{1}{2}$

Now we have the system. · $\,\mathrm{\{}\begin{array}{l} \,k=j-\frac{1}{2} \\ 65j=78k \\ \end{array}$ Solve the system of equations by substitution.

Substitute $k=j-\frac{1}{2}$ into the second equation, then solve for $j$.

$\,\begin{array}{rll} 65j & = & 78k \\ 65j & = & 78(j-\frac{1}{2}) \\ 65j & = & 78j-39 \\ -13j & = & -39 \\ j & = & 3 \\ \end{array}$

To find Kelly’s time, substitute $j=3$ into the first equation, then solve for $k$. · $\,k=j-\frac{1}{2}$

$\,k=3-\frac{1}{2}$

$\,k=\frac{5}{2}\,\mathrm{or}\,k=2\frac{1}{2}$

Check the answer in the problem.

Joni $\,3\,\mathrm{hours}\,(65\,\mathrm{mph})\,=195\,\mathrm{miles}$

Kelly $\,2\frac{1}{2}\,\mathrm{hours}\,(78\,\mathrm{mph})=195\,\mathrm{miles}$

Yes, they will have traveled the same distance when they meet.

Answer the question. · Kelly will catch up to Joni in $2\frac{1}{2}$ hours. By then, Joni will have traveled 3 hours.

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 is $b+c.$

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 4.22

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 4.23

Solution

Section 4.2 Exercises

Section 4.2 Exercises

Direct Translation Applications

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

Solve Geometry Applications

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

Solve Uniform Motion Applications

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

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

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

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?

1. The sum of two number is $-26.$ One number is 12 less than the other. Find the numbers.

2. The sum of two numbers is 37. Their difference is 9. Find the numbers.

3. The sum of two numbers is $-45.$ Their difference is $-89.$ Find the numbers.

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. June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?

1. Drew burned 1800 calories Friday playing one hour of basketball and canoeing for two hours. Saturday he spent two hours playing basketball and three hours canoeing and burned 3200 calories. How many calories did he burn per hour when playing basketball? How many calories did he burn per hour when canoeing?

2. Nancy bought seven pounds of oranges and three pounds of bananas for $17. Her husband later bought three pounds of oranges and six pounds of bananas for$12. What was the cost per pound of the oranges and the bananas?

3. Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for $40 or he is able to buy 5 packages of paper and 6 staplers for$62. How much does a package of paper cost? How much does a stapler cost?

1. The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

2. Owen is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make 100 quarts of lemonade?

3. The difference of two complementary angles is 17 degrees. Find the measures of the angles.

1. Two angles are complementary. The measure of the larger angle is ten more than four times the measure of the smaller angle. Find the measures of both angles.

2. The difference of two supplementary angles is 88 degrees. Find the measures of the angles.

3. Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.

1. 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.

2. 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.

3. Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his front yard. The length is five feet less than three times the width. Find the length and width of the fencing.

1. The perimeter of a toddler play area is 100 feet. The length is ten more than three times the width. Find the length and width of the play area.

2. College roommates John and David were driving home to the same town for the holidays. John drove 55 mph, and David, who left an hour later, drove 60 mph. How long will it take for David to catch up to John?

3. Felecia left her home to visit her daughter driving 45 mph. Her husband waited for the dog sitter to arrive and left home twenty minutes (1/3 hour) later. He drove 55 mph to catch up to Felecia. How long before he reaches her?

1. A motor boat travels 60 miles down a river in three hours but takes five hours to return upstream. Find the rate of the boat in still water and the rate of the current.

2. A river cruise boat sailed 80 miles down the Mississippi River for four hours. It took five hours to return. Find the rate of the cruise boat in still water and the rate of the current.

3. A small jet can fly 1435 miles in 5 hours with a tailwind but only 1,215 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

1. 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?

2. Translate to a system of equations and then solve:

Mark went to the gym and did 40 minutes of Bikram hot yoga and 10 minutes of jumping jacks. He burned 510 calories. The next time he went to the gym, he did 30 minutes of Bikram hot yoga and 20 minutes of jumping jacks burning 470 calories. How many calories were burned for each minute of yoga? How many calories were burned for each minute of jumping jacks?

3. Translate to a system of equations and then solve:

Erin spent 30 minutes on the rowing machine and 20 minutes lifting weights at the gym and burned 430 calories. During her next visit to the gym she spent 50 minutes on the rowing machine and 10 minutes lifting weights and burned 600 calories. How many calories did she burn for each minutes on the rowing machine? How many calories did she burn for each minute of weight lifting?

1. Translate to a system of equations and then solve:

The difference of two complementary angles is 20 degrees. Find the measures of the angles.

2. Translate to a system of equations and then solve:

The difference of two complementary angles is 80 degrees. Find the measures of the angles.

3. Translate to a system of equations and then solve:

Two angles are supplementary. The measure of the larger angle is 12 degrees more than three times the smaller angle. Find the measures of the angles.

1. Translate to a system of equations and then solve:

Two angles are supplementary. The measure of the larger angle is 18 less than twice the measure of the smaller angle. Find the measures of the angles.

2. 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.

3. 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.

1. Translate to a system of equations and then solve:

Mario wants to put a fence around the pool in his backyard. Since one side is adjacent to the house, he will only need to fence three sides. There are two long sides and the one shorter side is parallel to the house. He needs 155 feet of fencing to enclose the pool. The length of the long side is 10 feet less than twice the width. Find the length and width of the pool area to be enclosed.

2. Translate to a system of equations and then solve:

Alexis wants to build a rectangular dog run in her yard adjacent to her neighbor’s fence. She will use 136 feet of fencing to completely enclose the rectangular dog run. The length of the dog run along the neighbor’s fence will be 16 feet less than twice the width. Find the length and width of the dog run.

3. Translate to a system of equations and then solve:

Mitchell left Detroit on the interstate driving south towards Orlando at a speed of 60 miles per hour. Clark left Detroit 1 hour later traveling at a speed of 75 miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?

1. Translate to a system of equations and then solve:

Charlie left his mother’s house traveling at an average speed of 36 miles per hour. His sister Sally left 15 minutes $(\frac{1}{4}\,\mathrm{hour})$ later traveling the same route at an average speed of 42 miles per hour. How long before Sally catches up to Charlie?

2. Translate to a system of equations and then solve:

A Mississippi river boat cruise sailed 120 miles upstream for 12 hours and then took 10 hours to return to the dock. Find the speed of the river boat in still water and the speed of the river current.

3. Translate to a system of equations and then solve:

Jason paddled his canoe 24 miles upstream for 4 hours. It took him 3 hours to paddle back. Find the speed of the canoe in still water and the speed of the river current.

1. Translate to a system of equations and then solve:

A small jet can fly 1,325 miles in 5 hours with a tailwind but only 1,035 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

2. Translate to a system of equations and then solve:

A commercial jet can fly 1,728 miles in 4 hours with a tailwind but only 1,536 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

3. The sum of two number is 30. One number is 4 less than the other. Find the numbers.

1. A commercial jet can fly 1,320 miles in 3 hours with a tailwind but only 1170 miles in 3 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

2. Write a uniform motion problem similar to Example 4.15 that relates to where you live with your friends or family members. Then translate to a system of equations and solve it.