Slope-Intercept:

Point-Slope Form of a Line

We can also find the equation of a line when given the slope and any point (not the y-intercept), and there are two methods to do so. The following video will use a single example to show how to use both methods to find the equation of a line with a given slope and single point.

Video Source (09:40 mins) | Transcript

These are the two methods to finding the equation of a line when given a point and the slope:

Substitution method = plug in the slope and the (x, y) point values into y = mx + b, then solve for b. Use the m given in the problem, and the b that was just solved for, to create the equation y = mx + b.
Point-slope form = \({\text{y}} {-} {\text{y}}_1 = {\text{m}}({\text{x}}-{\text{x}}_1)\), where \(({\text{x}}_1, {\text{y}}_1)\) is the point given and m is the slope given. The 'x' and the 'y' stay as variables.

Additional Resources

Khan Academy: Intro to Point-Slope Form (06:07 mins, Transcript)
Khan Academy: Point-slope and Slope-intercept Equations (07:06 mins, Transcript)

Practice Problems

Find the equation of the line that passes through the point (1, 4) and has a slope of 12.

Find the equation of the line that passes through the point (1, 4) and has a slope of 2.

Find the equation of the line that passes through the point (27, 4) and has a slope of \(\frac{-2}{9}\).

Find the equation of the line that passes through the point \((-11, 2)\) and has a slope of \(\frac{-5}{11}\).

Find the equation of the line that passes through the point (10, 6) and has a slope of \(\frac{1}{5}\). What is the y-intercept of the line?

Find the equation of the line that passes through the point (3, 29) and has a slope of 6. What is the y-intercept of the line?

Solutions

\({\text{y}} = 12{\text{x}} - 8\)

\({\text{y}} = 2{\text{x}} + 2\) (Written Solution)

We will use the point-slope form to find the equation of the line. Point-slope form is:

\({\text{y}}{-}{\text{y}}_1={\text{m}}({\text{x}}-{\text{x}}_1)\)

We will use the point (1, 4) and the slope of 2 and substitute them in to \({\text{y}}{-}{\color{Purple}{\text{y}}_1}={\color{Orange}{\text{m}}}({\text{x}}-{\color{Red}{\text{x}}_1})\):

\({\text{y}}{-}{\color{Purple}4}={\color{Orange}2}({\text{x}}-{\color{Red}1})\)

Then distribute on the right side:

\({\text{y}}{-}{\color{Purple}4}={\color{Orange}2}{\text{x}}-{\color{Orange}2}(1)\)

Which simplifies to:

\({\text{y}}{-}{4}=2{\text{x}}{-}{2}\)

Add 4 to both sides:

\({\text{y}}{-}{4}{\color{Red}+4} = 2{\text{x}} - 2 {\color{Red}+4}\)

Which gives us the equation:

y = 2x +2

\({\text{y}} =-\frac{2}{9}{\text{x}}+10\)

\({\text{y}}=-\frac{5}{11}{\text{x}}-3\)

\(\left ( 0,4 \right )\) (Written Solution)

We will use the point-slope form to find the equation of the line. Point-slope form is:

\({\text{y}}{-}{\text{y}_{1}}=\text{m}\left ( {\text{x}}{-}{\text{x}_{1}} \right )\)

We will use the point (10, 6) and the slope of \({\color{DarkOrange} \frac{1}{5}}\) and substitute them in to:

\({\text{y}}{-}{\color{Purple} {\text{y}_{1}}}={\color{DarkOrange} \text{m}}\left ( {\text{x}}{-}{\color{Red} {\text{x}_{1}}} \right )\)

Then distribute on the right side:

\({\text{y}}{-}{\color{Purple}6}={\color{Orange}\frac{1}{5}}{\text{x}}-{\color{Orange}\frac{1}{5}}(10)\)

Which simplifies to:

\({\text{y}}{-}6={\color{Orange}\frac{1}{5}}{\text{x}}-2\)

Add 6 to both sides:

\({\text{y}}{-}6{\color{Red}+6}={\color{Orange}\frac{1}{5}}{\text{x}}-2{\color{Red}+6}\)

Which gives us the equation:

\(\text{y}={\color{DarkOrange} \frac{1}{5}}\text{x}+{\color{Red} 4}\)

The equation is in slope-intercept form, y=mx+b, so we can see that the y-intercept is at 4 or the point (0,4).

\((0,11)\)