Equation of a line: slope-intercept form

Objectives
Student will be able to:
  • write an equation in slope-intercept form when given the slope and y-intercept of a line.
  • write an equation in slope-intercept form when given the graph of an equation.
  • write an equation in slope-intercept form when given the slope and one point on a line.

Introduction
A hardware store rents drills to its customers. When a customer fails to return a drill by the return date, the hardware store charges a flat fee of $5 plus a fee of $2 per day.

This situation can be represented by an equation in slope-intercept form, , where y is dollars charged to a customer and x is the number of days late the drill is. The slope of the line is represented by m, and b is the y-intercept.



An equation in slope-intercept form allows:
  • all the values of y to be found (by substituting values for x and solving the equation) without plotting a graph.
  • ordered pairs to be found easily as the equation is already solved for y in terms of x.
  • the line of the equation to be graphed by finding two ordered pairs.
  • the line of the equation to be graphed by using the slope and y-intercept as identified in the equation.
Below is a graph illustrating the hardware store's fees for late drill returns. How much would Sara be charged if she is returning a drill two days late?



Read the graph to find that Sara would pay $9 if she returned a drill two days late. Solving the equation in slope-intercept form with the given value for x (2 days) supplies the same answer.



Write an equation of a line using slope-intercept form.

The equation of a line in slope-intercept form is:



        m is the slope                           b is the y-intercept

Write an equation in slope-intercept form of the line with a slope of 2 and a y-intercept of 4.

Step 1: Start by writing the values for the slope, m, and the y-intercept, b.

Step 2:
Write the equation of a line in slope-intercept form.

Step 3:
Substitute 2 for m and 4 for b.

The equation of the line is


Write an equation in slope-intercept form of the line on the graph.

Step 1:                
To find the value of the slope m, select two points on the line and use the slope formula or



Select two points on the line.


                                                            




OR



Apply the rise and run:

      

Step 2:
Find the value of the y-intercept, b, by observing where the line crosses the y-axis.

The y-intercept, b, is –1.
 
Step 3: Write the equation of a line in slope-intercept form.

Step 4:
Substitute   for m and –1 for b.

The equation of the line is

Write the equation of a line passing through (–3,0) with an undefined slope.

Step 1: Write the equation of a line with an undefined slope.

A line with an undefined slope is a vertical line. The equation of a vertical line is:

x = n (where n is a value of x at any point on the line)

Step 2:
Using the known point (–3,0), substitute the known value for x. 

The equation of the line is

Examples
1.
Write the equation of a line passing through (2,7) with an undefined slope.

Step 1:
Write the equation of a line with an undefined slope.

A line with an undefined slope is a vertical line. The equation of a vertical line is:

x = n (where n is a value of x at any point on the line)

Step 2:

Using the known point (2,7), substitute the known value for x. 
 
The equation of the line is x = 2.


2.
Write the equation of a horizontal line with a y-intercept of 6.

Step 1:
Start by writing the values for slope m and y-intercept b.

m = 0 because the slope of a horizontal line is zero
b = 6
 
Step 2:

Write the equation of a line in slope-intercept form.

Step 3:

Substitute 0 for m and 6 for b. Simplify.

The equation of the line is y = 6.


3.
Isaiah weighs 178 pounds now. He needs to lose weight to reach his ideal body mass index (BMI). Isaiah plans to lose weight at a rate of 1 pound per week. Write a linear equation to show Isaiah's weight after x weeks, assuming that he will lose one pound a week.

Step 1:
Write the linear form of an equation.

Step 2:

Substitute the values given in the problem.

The equation showing Isaiah's weight at x weeks is


Try Together

Complete the exercises. Show all work.

1.
Write the equation of the line on the graph.
 
Check student's work.
 

2.
Write the equation of the line with an undefined slope passing through the point (1,3).
 
Check student's work.
 

3.
Write the equation of the line with the given slope and y-intercept.
 

Check student's work.
 

4.
A taxi's fee consists of two parts. Jim's Taxi Service charges a flat fee of $6 and an additional $0.25 per mile. Write an equation to model the total charges y (in dollars) in terms of x, the number of miles driven.
 
 
Check student's work.
 

Please evaluate the student's answers:

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Good

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