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Research Foundations (c224) Linear Equation Word Problems Worksheets

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Western Governors University

Research Foundations (c224)

I. Model Problems
A linear model is a linear equation that represents a real-world
scenario. You can write the equation for a linear model in the same way
you would write the slope-intercept equation of a line. The y-intercept
of a linear model is the quantity that does not depend on x. The slope is
the quantity that changes at a constant rate as x changes. The change
must be at a constant rate in order for the equation to be a linear model.
Example 1 A machine salesperson earns a base salary of $40,000 plus
a commission of $300 for every machine he sells. Write an equation
that shows the total amount of income the salesperson earns, if he sells
x machines in a year.
The y-intercept is $40,000; the salesperson earns a $40,000 salary in a
year and that amount does not depend on x.
The slope is $300 because the salesperson’s income increases by $300
for each machine he sells.
Answer: The linear model representing the salesperson’s total income
is y = $300x + $40,000.
Linear models can be used to solve problems.
Example 2 The linear model that shows the total income for the
salesperson in example 1 is y = 300x + 40,000. (a) What would be the
salesperson’s income if he sold 150 machines? (b) How many
machines would the salesperson need to sell to earn a $100,000
income?
(a) If the salesperson were to sell 150 machines, let x = 150 in the linear
model; 300(150) + 40,000 = 85,000.
Answer: His income would be $85,000.
(b) To find the number of machines he needs to sell to earn a $100,000
income, let y = 100,000 and solve for x:
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y = 300x + 40,000 Write the linear model.
100,000 = 300x + 40,000 Substitute y = 100,000.
60,000 = 300x Subtract.
x = 200 Divide.
Answer: To earn a $100,000 income the salesperson would need to
sell 200 machines.
You can also use the standard form to write a linear model. Use this
form if you are analyzing two quantities that increase at different rates.
Example 3 At a school play, children’s tickets cost $3 each and adult
tickets cost $7 each. The total amount of money earned from ticket
sales equals $210. Write a linear model that relates the number of
children’s tickets sold to the number of adult tickets sold.

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Research Foundations (c224) Linear Equation Word Problems Worksheets

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