If you know the slope and any point on the line, you can write an equation of the line by using the slope formula. For example, suppose a line has a slope of 2 and contains (3, 5) . Let (x, y) be any other point on the line.
Slope Formula :
m = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute m = 2, (x_{1}, y_{1}) = (3, 5) and (x_{2}, y_{2}) = (x, y).
2 = (y - 5)/(x - 3)
Multiply each side by (x - 3).
2(x - 3) = y - 5
or
y - 5 = 2(x - 3)
The line with slope 'm' that contains the point (x_{1} , y_{1}) can be described by the equation
y - y_{1} = m(x - x_{1})
Write an equation in point-slope form for the line with the given slope that contains the given point.
Example 1 :
Slope = 5 ; (2, 0).
Solution :
Write the point-slope form.
y - y_{1} = m(x - x_{1})
Substitute 5 for m, 2 for x_{1} and 0 for y_{1}.
y - 0 = 5(x - 2)
Example 2 :
Slope = -7 ; (-2, 3).
Solution :
Write the point-slope form.
y - y_{1} = m(x - x_{1})
Substitute -7 for m, -2 for x_{1} and 3 for y_{1}.
y - 3 = -7[x - (-2)]
y - 3 = -7(x + 2)
A line can be graphed when given its equation in point-slope form. You can start by using the equation to identify a point on the line. Then use the slope of the line to identify a second point.
Graph the line described by each equation.
Example 3 :
y - 1 = 3(x - 1)
Solution :
y - 1 = 3 (x - 1) is in the form y - y_{1} = m(x - x_{1}).
Slope m = 3 = 3/1
The line contains the point (1, 1) .
Step 1 :
Plot (1, 1).
Step 2 :
Count 3 units up and 1 unit right and plot another point.
Step 3 :
Draw the line connecting the two points.
Example 4 :
y + 2 = (-1/2)(x - 3)
Solution :
Step 1 :
Write the equation in point-slope form :
y - y_{1} = m(x - x_{1})
y + 2 = (-1/2)(x - 3)
Rewrite addition of 2 as subtraction of -2.
y - (-2) = (-1/2)(x - 3)
Step 2 :
The line contains the point (3, -2).
Slope m = -1/2 = 1/(-2)
Write the equation that describes each line in slope-intercept form.
Example 5 :
slope = -4, (-1, -2) is on the line.
Solution :
Because the slope of the line and a point on the line are given, we can write the equation of the line in point-slope form.
y - y_{1} = m(x - x_{1})
Substitute m = -4 and (x_{1}, y_{1}) = (-1, -2).
y - (-2) = -4[x - (-1)]
Simplify and solve for y :
y + 2 = -4(x + 1)
Distribute -4 on the right side.
y + 2 = -4x - 4
Subtract 2 from each side.
y + 2 = -4x - 4
y = -4x - 6
Example 6 :
(1, -4) and (3, 2) are on the line.
Solution :
Find the slope.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
= [2 - (-4)]/(3 - 1)
= (2 + 4)/2
= 6/2
= 3
Substitute the slope and one of the points into the point-slope form. Then write the equation in slope-intercept form.
y - y_{1} = m(x - x_{1})
Substitute m = 3, (x_{1}, y_{1}) = (3, 2).
y - 2 = 3(x - 3)
Simplify.
y - 2 = 3x - 9
Add 2 to each side.
y = 3x - 7
Example 7 :
x-intercept = –2, y-intercept = 4.
Solution :
Use the intercepts to find two points :
(-2, 0) and (0, 4)
Find the slope.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
= (4 - 0)/[(0 -(-2)]
= 4/2
= 2
Write the equation in slope-intercept form.
y = mx + b
Substitute 2 for m and 4 for b.
y = 2x + 4
Example 8 :
The points (4, 8) and (-1, -12) are on a line. Find the intercepts.
Solution :
Step 1 :
Find the slope.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
= (-12 - 8)/(-1 - 4)
= -20/(-5)
= 4
Step 2 :
Write the equation in point-slope form.
y - y_{1} = m(x - x_{1})
Substitute m = 4, (x_{1}, y_{1}) = (4, 8).
y - 8 = 4(x - 4)
Simplify and solve for y.
y - 8 = 4x - 16
Add 8 to each side.
y = 4x - 8
Step 3 :
Find the intercepts :
x - intercept :
0 = 4x - 8
8 = 4x
2 = x
y - intercept :
y = 4(0) - 8
y = -8
y = -8
The x-intercept is 2, and the y-intercept is -8.
Example 9 :
The cost to place an ad in a newspaper for one week is a linear function of the number of lines in the ad. The costs for 3, 5, and 10 lines are shown. Write an equation in slope-intercept form that represents the function. Then find the cost of an ad that is 18 lines long.
Solution :
Understand the Problem :
• The answer will have two parts—an equation in slope-intercept form and the cost of an ad that is 18 lines long.
• The ordered pairs given in the table satisfy the equation.
Make a Plan :
First, find the slope. Then use point-slope form to write the equation. Finally, write the equation in slope-intercept form.
Solve :
First, find the slope. Then use point-slope form to write the equation. Finally, write the equation in slope-intercept form.
Step 1 :
Choose any two ordered pairs from the table to find the slope.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
Use (3, 13.50) and (5, 18.50).
= (18.50 - 13.50)/(5 - 3)
= 5/2
= 2.5
Step 2 :
Substitute the slope and any ordered pair from the table into the point-slope form.
y - y_{1} = m(x - x_{1})
Substitute m = 2.5, (x_{1}, y_{1}) = (10, 31).
y - 31 = 2.5(x - 10)
Step 3 :
Write the equation in slope-intercept form by solving for y.
y - 31 = 2.5(x - 10)
Distribute 2.5.
y - 31 = 2.5x - 25
Add 31 to each side.
y = 2.5x + 6
Step 4 :
Find the cost of an ad containing 18 lines by substituting 18 for x.
y = 2.5x + 6
y = 2.5(18) + 6
y = 45 + 6
y = 51
The cost of an ad containing 18 lines is $51.
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