Point-Slope

Point-Slope Form of the Equation of a Line
\bf\displaystyle 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 (provided by Mathematics Formula Sheet)
point-slope form of the equation of a line
\bf\displaystyle y-{{y}_{1}}=m(x-{{x}_{1}})

slope-intercept form of the equation of a line
\bf\displaystyle 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
\bf\displaystyle y-{{y}_{1}}=m(x-{{x}_{1}})
(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
\bf\displaystyle 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
\bf\displaystyle y-{{y}_{1}}=m(x-{{x}_{1}})
(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
\bf\displaystyle 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
\bf\displaystyle y-{{y}_{1}}=m(x-{{x}_{1}})
(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
\bf\displaystyle 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

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