You can graph a straight line using its slope and the point that contains the 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 = (y_{2} - y_{1})/(x_{2} - x_{1})
3 = (y - 5)/(x - 0)
Because we do not know (x_{2}, y_{2}), 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
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.
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 = (y_{2} - y_{1}) / (x_{2} - x_{1})
Substitute (0, 1) for (x_{1} , y_{1}) and (1, 3) for (x_{2} , y_{2}).
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
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.
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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