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Problem Solving using the X-Intercept

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to solve for the x-intercept and use the x-intercept to solve for real world problems. As you'll see, practicing x-intercept equations strengthens your problem solving skill in your personal and professional life. Specifically, this lesson covers:

Table of Contents

1. Problem Solving Using the x-Intercept

We have talked quite a bit about the y-intercept of a line. But what about the x-intercept? The x-intercept is the location where the line crosses the x-axis, and it has coordinates of (x, 0). Well, we can also use the equation in this form to find the x-intercept. What is the value of y across the x-axis? It’s 0! Since x-intercepts have a y-coordinate of 0, we set the equation equal to 0 and solve for x.

EXAMPLE

The equation below represents the balance of debt during a 24-month repayment period, with the corresponding graph representation.

y equals short dash 45 x plus 1080

A graph titled ‘Debt Balance’ with the x-axis labeled ‘Time (months)’ and ranging from 0 to 24 and the y-axis labeled ‘Balance (Dollars)’ and ranging from 0 to 1,200. A dotted line slants downward from left to right, starting between 1,000 and 1,200 dollars on the y-axis and ending at 24 months on the x-axis.

To find the x-intercept, we substitute y for 0 in the linear equation.

y equals short dash 45 x plus 1080 The equation
0 equals short dash 45 x plus 1080 To find the x-intercept, plug 0 in for y as the y-coordinate is always 0 when the line crosses the x-axis.
short dash 1080 equals short dash 45 x Solve the equation by subtracting 1080 from both sides.
24 equals x Divide both sides by -45.
left parenthesis 24 comma space 0 right parenthesis The x-intercept

We can also see this in the graph. According to our calculations and chart, it will take 24 months to pay off the balance of $1080.

try it
Suppose you have the equation y equals short dash 2 x plus 8.
What is the x-intercept?
y equals short dash 2 x plus 8 The equation.
0 equals short dash 2 x plus 8 To find the x-intercept, plug 0 in for y as the y-coordinate is always 0 when the line crosses the x-axis.
short dash 8 equals short dash 2 x Solve the equation by subtracting 8 from both sides.
4 equals x Divide both sides by -2.
left parenthesis 4 comma space 0 right parenthesis This is the x-coordinate of the x-intercept.

For the equation y equals short dash 2 x plus 8, the x-intercept is at (4, 0).

try it
Suppose you have the equation y equals 5 x minus 30.
What is the x-intercept?
y equals 5 x minus 30 The equation.
0 equals 5 x minus 30 To find the x-intercept, plug 0 in for y as the y-coordinate is always 0 when the line crosses the x-axis.
30 equals 5 x Solve the equation by adding 30 to both sides.
6 equals x Divide both sides by 5.
left parenthesis 6 comma space 0 right parenthesis Our x-intercept

term to know
X-Intercept
The location on a graph where a line or curve intersects the x-axis: (x, 0).

2. Using the x-Intercept to Solve Real World Problems

The equation below represents the balance of a $2000 debt with no interest added. If $250 is paid each month, how long will it take to pay back the loan (or for the balance to be 0)? The equation can be written as: y equals short dash 250 x plus 2000. Solve the problem by solving for the x-intercept.

y equals short dash 250 x plus 2000 The equation.
0 equals short dash 250 x plus 2000 To find the x-intercept, plug 0 in for y as the y-coordinate is always 0 when the line crosses the x-axis.
short dash 2000 equals short dash 250 x Solve the equation by subtracting 2000 from both sides.
8 equals x Divide both sides by -250.
left parenthesis 8 comma space 0 right parenthesis Our x-intercept

The x-intercept is (8, 0). This means that the loan will be paid back in 8 months.

Problem Solving: Skill Reflect
The x-intercept is frequently used to solve problems in which predictions are made. You can use the x-intercept to determine when you will pay off your car or to estimate your company's sales for the next year. Think about problems or predictions that you can solve by using the x-intercept.

summary
In this lesson, you learned how to solve problems using the x-intercept, which is the location where the line crosses the x-axis, represented by the coordinates of (x, 0). Since x-intercepts have a y-coordinate of 0, you can use the slope-intercept form to find the x-intercept by setting the equation equal to 0 and solving for x. You also learned how to use the x-intercept to solve real world problems, such as determining how long it will take to pay off a loan.

Best of luck in your learning!

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