**Point-Slope Form of the Equation of a Line**

point-slope form of the equation of a line

y – y_{1} = m(x – x_{1})

**
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 – y_{1} = m(x – x_{1})

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 – y_{1} = m(x – x_{1})

(x_{1}, y_{1}) = (-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 – y_{1} = m(x – x_{1})

(x_{1}, y_{1}) = (-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 – y_{1} = m(x – x_{1})

(x_{1}, y_{1}) = (-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