# SLOPE INTERCEPT FORM

You can graph a straight line using its slope and the point that contains the y-intercept.

## Graphing by Using Slope and y-intercept

Example 1 :

Graph the line with slope -2 and y-intercept 4.

Solution :

Step 1 :

The y-intercept is 4, so the line contains (0, 4). Plot (0, 4).

Step 2 :

Slope  =  change in y / change in x  =  -2/1

Count 2 units down and 1 unit right from (0, 4) and plot another point.

Step 3 :

Draw the line through the two points. If you know the slope of a line and the y-intercept, you can write an equation that describes the line.

Step 1 :

If a line has slope 2 and the y-intercept is 3, then m = 3 and (0, 5) is on the line. Substitute these values into the slope formula.

Slope formula :

m  =  (y2 - y1)/(x2 - x1)

3  =  (y - 5)/(x - 0)

Because we do not know (x2, y2), we use (x, y).

Step 2 :

Solve for y :

3  =  (y - 5)/(x - 0)

3  =  (y - 5)/x

Multiply each side by x.

3x  =  y - 5

Add 5 to each side.

3x + 5  =  y

or

y  =  3x + 5

## Slope-Intercept Form of a Linear Equation

If a line has slope m and the y-intercept is b, then the line is described by the equation

y  =  m x + b

Any linear equation can be written in slope-intercept form by solving for y and simplifying.

In this form, you can immediately see the slope and y-intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

## Writing Linear Equations in Slope-Intercept Form

Write the equation that describes each line in slope-intercept form.

Example 2 :

Slope = 2/5, y-intercept = 6.

Solution :

Write the slope-intercept form equation of a line :

y  =  mx + b

Substitute 2/5 for m and 6 for b.

y  =  (2/5)x + 6

y  =  2x/5 + 6

Example 3 :

Slope = 0, y-intercept = -4.

Solution :

Write the slope-intercept form equation of a line :

y  =  mx + b

Substitute 0 for m and -4 for b.

y  =  (0)x + (-4)

y  =  0 - 4

y  =  -4

Example 4 : Solution :

Step 1 :

Find the y-intercept.

The graph crosses the y-axis at (0, 1), so b = 1.

Step 2 :

Find the slope.

The line contains the points (0, 1) and (1, 3).

Use the slope formula.

m  =  (y2 - y1) / (x2 - x1)

Substitute (0, 1) for (x1 , y1and (1, 3) for (x2 , y2).

m  =  (3 - 1) / (1 - 0)

m  =  2/1

m  =  2

Step 3 :

Write the slope-intercept form equation of a line :

y  =  mx + b

Substitute 2 for m and 1 for b.

y  =  2x + 1

Example 5 :

Slope = 4, (2, 5) is on the line.

Solution :

Step 1 :

Find the y-intercept.

Write the slope-intercept form equation of a line :

y  =  mx + b

Substitute 4 for m, 2 for x, and 5 for y.

5  =  4(2) + b

5  =  8 + b

Subtract 8 from each side.

-3  =  b

Step 2 :

Write the slope-intercept form equation of a line :

y  =  mx + b

Substitute 4 for m and -3 for b.

y  =  4x + (-3)

y  =  4x - 3

## Using Slope-Intercept Form to Graph

Write each equation in slope-intercept form. Then graph the line described by the equation.

Example 6 :

4x - y - 3  =  0

Solution :

Step 1 :

Write the given equation slope-intercept form :

4x - y - 3  =  0

Add y to each side.

4x - 3  =  y

y  =  4x - 3

y = 4x - 3 is in the form y = mx + b.

Slope : m  =  4  =  4/1

y-intercept : b  =  -3

Step 2 :

Plot (0, -3).

Step 3 :

Count 4 units up and 1 unit right and plot another point.

Step 4 :

Draw the line connecting the two points. Example 7 :

2x + 3y - 6  =  0

Solution :

Step 1 :

Write the given equation slope-intercept form :

2x + 3y - 6  =  0

Subtract 2x from each side and add 6 to each side.

3y  =  -2x + 6

Divide each side 3.

3y/3  =  (-2x + 6)/3

y  =  -2x/3 + 6/3

y  =  (-2/3)x + 2

Slope : m  =  -2/3

y-intercept : b  =  2

Step 2 :

Plot (0, 2).

Step 3 :

Count 2 units down and 3 unit right and plot another point.

Step 4 :

Draw the line connecting the two points. Example 8 :

3x + 2y  =  8

Solution :

Step 1 :

Write the given equation slope-intercept form :

3x + 2y  =  8

Subtract 3x from each side and add 6 to each side.

2y  =  -3x + 8

Divide each side 2.

2y/2  =  (-3x + 8)/2

y  =  -3x/2 + 8/2

y  =  (-3/2)x + 4

Slope : m  =  -3/2

y-intercept : b  =  4

Step 2 :

Plot (0, 4).

Step 3 :

Count 3 units down and 2 unit right and plot another point.

Step 4 :

Draw the line connecting the two points. ## Consumer Application

Example 9 :

To rent a vehicle, a moving company charges \$30.00 plus \$0.50 per mile. The cost as a function of the number of miles driven is shown in the graph. (i) Write an equation that represents the cost as a function of the number of miles.

(ii) Identify the slope and y-intercept and describe their meanings.

(iii) Find the cost of the van for 150 miles.

Solution :

Part (i) :

Cost is \$0.50 per mile times miles plus \$30.00.

y  =  0.50x + 30

An equation is y = 0.5x + 30.

Part (ii) :

The y-intercept is 30. This is the cost for 0 miles, or the initial fee of \$30.00.

The slope is 0.5.

This is the rate of change of the cost : \$0.50 per mile.

Part (iii) :

y  =  0.5x + 30

Substitute 150 for x in the equation.

=  0.5(150) + 30

=  75 + 30

=  105

The cost of the vehicle for 150 miles is \$105.

Example 10 :

A caterer charges a \$200 fee plus \$18 per person served. The cost as a function of the number of guests is shown in the graph. (i) Write an equation that represents the cost as a function of the number of guests.

(ii) Identify the slope and y-intercept and describe their meanings.

(iii) Find the cost of catering an event for 200 guests.

Solution :

Part (i) :

Cost is \$18 per guest plus \$200.

y  =  18x + 200

An equation is y = 18x + 200.

Part (ii) :

The y-intercept is 200. This is the cost for 0 guests, or the initial fee of \$200.

The slope is 18.

This is the rate of change of the cost : \$18 per guest.

Part (iii) :

y  =  18x + 50

Substitute 200 for x in the equation.

=  18(200) + 200

=  3600 + 200

=  3800

The cost of catering for 200 guests is \$3800.

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