Systems of Equations – Word Problems – Practice Problems

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We often run into problems where we have multiple answers to find, and where the answers depend on each other. For instance, if we want to figure out what temperature it feels like outside, or the heat index, we would need to figure out both the air temperature and how humid it is outside.

Such problems, when expressed mathematically, end up givng a system of equations with multiple variables to solve for. Let’s look at a particular example: say your younger cousins worked part-time jobs over the summer to save money for a car. The job paid different rates for working weekdays and weekends. During the first week, one cousin worked 22 hours during the week and one hour on the weekend, earning $232. Her sister earned $270 by working 15 hours during the week and 10 hours on the weekend. What was the hourly pay rate for working weekdays and weekends?

When solving word problems, it is important to understand what you are asked to find. This problem is asking us to find the hourly pay rate for weekdays and the hourly pay rate for weekends. We will need to use different variables to represent each unknown. Let's use x for the weekday rate and y for the weekend rate. Now we can use the other information from the word problem to create equations that we will use to solve for the variables.

One cousin worked 22 weekday hours and one weekend hour for a total of $232. We can represent this with the equation 22x + y = 232. Using the pay for the other cousin, we get the equation 15x + 10y = 270. Since there are two equations with the same two variables, we can set them up as a system of equations and solve them simultaneously.
22x + y = 232
15x + 10y = 270

Here, we use substitution method to solve the system. The first equation can be solved for y by subtracting 22x from both sides of the equation. This leaves us with y=232 - 22x. Since y is equal to the expression 232 - 22x, we can substitute the expression for y in the second equation. The equation becomes 15x+ 10(232-22x) = 270. We now have one equation with one variable to solve.

15x + 10(232 - 22x) = 270
15x + 2320 - 220x = 270
2320 - 205x = 270
- 205x = -2 050
x = 10

We now know that the hourly pay rate for weekdays is $10. By substituting 10 for x in one of the equations, we can solve for y. Using the revised version of the first equation, we get
y = 232 - 22(10) = 12.

Now that the values of both variables are known, the word problem is solved. The hourly pay rate is $10 for weekdays and $12 for weekends.

Solve Systems of Equations.

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Exercises in this Practice Problem
Determine the costs of supplies for cats as well as dogs.
Solve for the number of cats Jessica could buy supplies for.
Describe how to solve a system of equations by graphing.
Determine the price of dog as well as cat supplies.
Explain how to solve systems of equations.
Decide which graph belongs to the system of equations.