Table of Contents
- Equation of a Line: Given Slope and One Point
- Example 1: Determine the equation of a line with a slope of 2 passing through the point (1, 4).
- Example 2: Determine the equation of the line with a slope of 2/3 passing through (6, 7).
- Example 3: Find the equation of the line parallel to y = 3x 2 passing through the point (2, 5).
- Example 4: Find the equation of the line perpendicular to 2x-3y + 7 = 0 with the same yintercept as 4x + 5y 15 = 0.
Equation of a Line: Given Slope and One Point
Example 1: Determine the equation of a line with a slope of 2 passing through the point (1, 4).
Solution: m=2, (-1,4)
Write the equation of the line in slope y-intercept form,
y=mx+b Write the form you’re using.
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y=2x+b Substitute the slope into m.
-Sub. x = – 1 and y = 4 into (*) and solve for b.
y = 2x + b
4 = 2(- 1) + b Substitute the coordinates into x and y.
4 = – 2 + b Simplify.
4+2 = b Isolate the ‘b’ term
6 = b Simplify
b = 6 Rearrange so the variable is on the left.
-Now substitute m = 2 and b = 6 in y = mx+b.
∴y = 2x + 6 This is the equation of the line with a slope of 2 possing through the point (-1,4).
Example 2: Determine the equation of the line with a slope of 2/3 passing through (6, 7).
Solution: m=23, (6, -7)
y = mx + b ← Write the form you’re using.
y = 23x + b ← Substitute the slope into m.
– 7 = 23(6) + b ← Substitute the coordinates into x and y.
– 7 = (2)(2) + b ← Simplify.
– 7 = 4 + b ← Isolate the ‘b’ term
– 7 – 4 =b ← Simplify
– 11 = b ← Simplify
b = – 11 Rearrange so the variable is on the left.
-Sub. m=23 and b = – 11 into y= mx+b
Write the equation
∴ y=23 x – 11
Equation of the line that has a slope of 23 and passes through
(6,-7).
Example 3: Find the equation of the line parallel to y = 3x 2 passing through the point (2, 5).
Solution: y = 3x – 2, m = 3
-Since the line whose equation we are trying to write is parallel to y = 3x – 2, then the slope of this line is 3.
-Use m = 3 and (-2, 5) to write the equation of the line.
y = mx + b y= 3x +11
Y = 3x + b
-Use (-2, 5) to find b
5 = 3(- 2) + b
5 = – 6 + b
5+6 = b
11 = b
b=11
– Sub . m = 3 and b = 11 into y = mx + b
∴ y = 3x + 11 Equation of the line that is parallel to y = 3x – 2 passing through (-2, 5)
Example 4: Find the equation of the line perpendicular to 2x-3y + 7 = 0 with the same yintercept as 4x + 5y 15 = 0.
Solution:
Write 2x-3y +7=0 in slope y-intercept form, y = mx + b
2x + 7 = 3y
3y = 2x + 7
3y3=2x+73
y=23x+73
So, m=23
The slope of the line perpendicular to
2x – 3y + 7 = 0 is -32 ( recall : m1=-1m2)
-Next, find the y-intercept of 4x+5y-15=0
Set x=0,
As a point (0,3)
-Write the equation of the line using the slope of -32 and point (0,3).
∴ y=23 x +3 is the equation of the – line that is perpendicular to 2x – 3y + 7 = 3 with the same y-intercept as 4x+5y-15=0
