4. Systems of Equations
4.4. Solve Applications with Systems of Equations
Lynn Marecek and MaryAnne Anthony-Smith
Learning Objectives
By the end of this section it is expected that you will be able to:
- Translate to a system of equations
- Solve direct translation applications
- Solve geometry applications
- Solve uniform motion applications
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.
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose variables to represent those quantities.
- Translate into a system of equations.
- Solve the system of equations using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- 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.
Let’s see how we can translate these problems into a system of equations with two variables. We’ll focus on Steps 1 through 4 of our Problem Solving Strategy.
EXAMPLE 1
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.
TRY IT 1
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.
Show answer
We’ll do another example where we stop after we write the system of equations.
EXAMPLE 2
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?
| We are looking for the amount that the husband and wife each earn. | Let the amount the husband earns. the amount the wife earns. |
| Translate. | A married couple together earns $110,000. |
| The wife earns $16,000 less than twice what husband earns. | |
| The system of equations is: |
TRY IT 2
Translate to a system of equations:
A couple has a total household income of $84,000. The husband earns $18,000 less than twice what the wife earns. How much does the wife earn?
Show answer
Solve Direct Translation Applications
We set up, but did not solve, the systems of equations in examples 1 and 2. Now we’ll translate a situation to a system of equations and then solve it.
EXAMPLE 3
Translate to a system of equations and then solve:
Devon is 26 years older than his son Cooper. The sum of their ages is 50. Find their ages.
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | We are looking for the ages of Devon and Cooper. |