Translating 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 tutorial 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 tutorial 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.

Example: 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.”

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

Example

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

$\begin{array}{cccc}\text{We are looking for the amount that}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}h=\text{the amount the husband earns.}\hfill \\ \text{the husband and wife each earn.}\hfill & & & \phantom{\rule{1.2em}{0ex}}w=\text{the amount the wife earns}\hfill \\ \\ \text{Translate.}\hfill & & & \text{A married couple together earns \$110,000.}\hfill \\ & & & \hfill w+h=110,000\hfill \\ & & & \begin{array}{c}\text{The wife earns \$16,000 less than twice what}\hfill \\ \text{husband earns.}\hfill \end{array}\hfill \\ & & & \hfill w=2h-16,000\hfill \\ \\ \text{The system of equations is:}\hfill & & & \hfill \begin{array}{c}w+h=110,000\hfill \\ w=2h-16,000\hfill \end{array}\hfill \end{array}$

This lesson is part of:

Systems of Linear Equations I

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