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5.4 Solve applications with systems of equations

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By the end of this section, you will be able to:
  • Translate to a system of equations
  • Solve direct translation applications
  • Solve geometry applications
  • Solve uniform motion applications

Before you get started, take this readiness quiz.

  1. The sum of twice a number and nine is 31. Find the number.
    If you missed this problem, review [link] .
  2. Twins Jon and Ron together earned $96,000 last year. Ron earned $8,000 more than three times what Jon earned. How much did each of the twins earn?
    If you missed this problem, review [link] .
  3. Alessio rides his bike 3 1 2 hours at a rate of 10 miles per hour. How far did he ride?
    If you missed this problem, review [link] .

Previously in this chapter we solved several applications with systems of linear equations. In this section, we’ll look at some specific types of applications that relate two quantities. We’ll translate the words into linear equations, decide which is the most convenient method to use, and then solve them.

We will use our Problem Solving Strategy for Systems of Linear Equations.

Use a problem solving strategy for systems of linear 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.

Translate to a system of equations

Many of the problems we solved in earlier applications related two quantities. Here are two of the examples from the chapter on Math Models .

  • The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
  • A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

In that chapter we translated each situation into one equation using only one variable. Sometimes it was a bit of a challenge figuring out how to name the two quantities, wasn’t it?

Let’s see how we can translate these two problems into a system of equations with two variables. We’ll focus on Steps 1 through 4 of our Problem Solving Strategy.

How to translate to a system of equations

Translate to a system of equations:

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

Solution

This figure has four rows and three columns. The first row reads, “Step 1: Read the problem. Make sure you understand all the words and ideas. This is a number problem. The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.” The second row reads, “Step 2: Identify what you are looking for. ‘Find the numbers.’ We are looking for 2 numbers.” The third row reads, “Step 3: Name what you are looking for. Choose variables to represent those quantities. We will use two variables, m and n. Let me = one number n = second number.” The fourth row reads, “Step 4: Translate into a system of equations. We will write one equation for each sentence.” The figure then shows how, “The sum of the numbers is -14” becomes m + n = -14 and “One number is four less than the other” becomes m = n – 4. The figure then says, “The system is m + n = -14 and m = n – 4.”
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Translate to a system of equations:

The sum of two numbers is negative twenty-three. One number is 7 less than the other. Find the numbers.

{ m + n = −23 m = n − 7

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Translate to a system of equations:

The sum of two numbers is negative eighteen. One number is 40 more than the other. Find the numbers.

{ m + n = −18 m = n + 40

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We’ll do another example where we stop after we write the system of equations.

Translate to a system of equations:

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

Solution

We are looking for the amount that Let h = the amount the husband earns. the husband and wife each earn. w = the amount the wife earns Translate. A married couple together earns $110,000. w + h = 110,000 The wife earns $16,000 less than twice what husband earns. w = 2 h − 16,000 The system of equations is: { w + h = 110,000 w = 2 h − 16,000

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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