Point-Slope Form of the Equation of a Line
point-slope form of the equation of a line
y – y1 = m(x – x1)
Basics
The point-slope form of the equation of a line is a way to construct the equation of a line without graphing.
A question involving point-slope could ask you to use one point and the slope to write the point-slope equation of a line.
You could also be asked to convert the equation of a line from point-slope to slope-intercept form.
In order to convert the equation of a line from point-slope to slope-intercept form, it is helpful to understand algebraic equations.
Formulas (Mathematics Formula Sheet)
point-slope form of the equation of a line
y – y1 = m(x – x1)
slope-intercept form of the equation of a line
y = mx + b
Question
Write the point-slope form of the equation of a line passing through point (-2, 4) with a slope of 5.
Answer
y – 4 = 5(x + 2)
Answer Process
point-slope form of the equation of a line
y – y1 = m(x – x1)
(x1, y1) = (-2, 4)
m = 5
y – 4 = 5(x – -2)
y – 4 = 5(x + 2)
• two consecutive negative signs produce a positive sign (see negatives)
point-slope form of the equation of a line
y – 4 = 5(x + 2)
Question
With reference to the question above, convert the equation of the line from point-slope to slope-intercept form.
Answer
y = 5x + 14
Answer Process
slope-intercept form of the equation of a line
y = mx + b
Proceed from the point-slope form.
y – 4 = 5(x + 2)
y – 4 = 5·x + 5·2
• There is an invisible multiplication sign between 5 and both components of (x + 2).
y – 4 = 5x + 10
• multiplication processed
y – 4 + 4 = 5x + 10 + 4
• 4 is added to each side of = sign
• addition is the opposite of subtraction
y = 5x + 14
• -4 + 4 is the same as 0, so it disappears
• 10 + 4 is the same as 14
• y has been isolated on the left side of = sign
slope-intercept form of the equation of a line
y = 5x + 14
Question
Write the point-slope form of the equation of a line passing through point (-3, -6) with a slope of -7.
Answer
y + 6 = -7(x + 3)
Answer Process
point-slope form of the equation of a line
y – y1 = m(x – x1)
(x1, y1) = (-3, -6)
m = -7
y – -6 = -7(x – -3)
y + 6 = -7(x + 3)
• two consecutive negative signs produce a positive sign (see negatives)
point-slope form of the equation of a line
y + 6 = -7(x + 3)
Question
With reference to the question above, convert the equation of the line from point-slope to slope-intercept form.
Answer
y = -7x – 27
Answer Process
slope-intercept form of the equation of a line
y = mx + b
Proceed from the point-slope form.
y + 6 = -7(x + 3)
y + 6 = -7·x + -7·3
• There is an invisible multiplication sign between -7 and both components of (x + 3).
y + 6 = -7x + -21
• multiplication processed
y + 6 = -7x – 21
• + -21 is the same as – 21 (see negatives)
y + 6 – 6 = -7x – 21 – 6
• 6 is subtracted from each side of = sign
• subtraction is the opposite of addition
y = -7x – 27
• 6 – 6 is the same as 0, so it disappears
• -21 – 6 is the same as -27
• y has been isolated on the left side of = sign
slope-intercept form of the equation of a line
y = -7x – 27
Question
Write the slope-intercept form of the equation of a line passing through point (-4, 9) with a slope of 2.
Answer
y = 2x + 17
Answer Process
point-slope form of the equation of a line
y – y1 = m(x – x1)
(x1, y1) = (-4, 9)
m = 2
y – 9 = 2(x – -4)
y – 9 = 2(x + 4)
• two consecutive negative signs produce a positive sign (see negatives)
point-slope form of the equation of a line
y – 9 = 2(x + 4)
slope-intercept form of the equation of a line
y = mx + b
Proceed from the point-slope form.
y – 9 = 2(x + 4)
y – 9 = 2·x + 2·4
• There is an invisible multiplication sign between 5 and both components of (x – 2).
y – 9 = 2x + 8
• multiplication processed
y – 9 + 9 = 2x + 8 + 9
• 9 is added to each side of = sign
• addition is the opposite of subtraction
y = 2x + 17
• -9 + 9 is the same as 0, so it disappears
• 8 + 9 is the same as 17
• y has been isolated on the left side of = sign
slope-intercept form of the equation of a line
y = 2x + 17
Practice – Questions
1. Write the point-slope form of the equation of a line passing through point (-2, 5) with a slope of 4.
2. With reference to Question 1, convert the equation of the line from point-slope to slope-intercept form.
3. Write the point-slope form of the equation of a line passing through point (-5, -6) with a slope of -9.
4. With reference to Question 3, convert the equation of the line from point-slope to slope-intercept form.
5. Write the slope-intercept form of the equation of a line passing through point (-4, 8) with a slope of 6.
Practice – Answers
1. y – 5 = 4(x + 2)
2. y = 4x + 13
3. y + 6 = -9(x + 5)
4. y = -9x – 51
5. y = 6x + 32