Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We'll modify the strategy slightly here to make it appropriate for systems of equations.

HOW TO

Use a problem solving strategy for systems of linear equations.

  1. Step 1. Read the problem. Make sure all the words and ideas are understood.

  2. Step 2. Identify what we are looking for.

  3. Step 3. Name what we are looking for. Choose variables to represent those quantities.

  4. Step 4. Translate into a system of equations.

  5. Step 5. Solve the system of equations using good algebra techniques.

  6. Step 6. Check the answer in the problem and make sure it makes sense.

  7. Step 7. Answer the question with a complete sentence.

Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

Example 5.12

Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?

Solution

Step 1. Read the problem.

Step 2. Identify what we are looking for.

We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.

Step 3. Name what we are looking for. Choose variables to represent those quantities.

  Let f= number of quarts of fruit juice.
    c= number of quarts of club soda

Step 4. Translate into a system of equations.

\( \underbrace {\text{The number of quarts of fruit juice}}_{f} \, \underbrace{\text{and }}_{+} \, \underbrace{\text{the number of quarts of club soda}}_{c} \, \underbrace{\text{is}} \; _{=} \underbrace{10}_{10} \)

\( \underbrace{\text{The number of  quarts of fruit juice}}_{f} \, \underbrace{\text{is}}_{=} \; \underbrace{\text{four times the number of soda}}_{4c} \)

We now have the system. \(\left\{\begin{array}{l}f+c=10 \\ f=4 c\end{array}\right.\)

Step 5. Solve the system of equations using good algebra techniques.

The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.

Step 6. Check the answer in the problem and make sure it makes sense.

Does this make sense in the problem?

Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.

Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.

Step 7. Answer the question with a complete sentence.

Sondra needs 8 quarts of fruit juice and 2 quarts of soda.

Try It 5.23

Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Try It 5.24

Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

Back to '6.2: Solving Systems of Linear Equations by Graphing'