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: