Introduction

The point-slope form of a line that passes through any point (x_1,y_1) with slope m is given by

y - y_1 = m(x-x_1).

For example, consider the straight line graph which passes through the points (3,1) and (5,5) , as shown below.

We know that the line passes through the point (3,1), so we can use (x_1,y_1) = (3,1) in our point-slope formula. All that remains is to compute m, and we can do this using the given points.

First, calculate the slope m using the given points: \begin{align} m = \frac{y_2 - y_1}{x_2 - x_1} =\frac{5 - 1}{5 - 3} =\frac{4}{2} =2 \end{align}

Then, we substitute m=2 and the point (x_1,y_1) = (3,1) into the point-slope formula:

\begin{align} y - y_1 &= m(x-x_1) \\[3pt] y-1 &=2(x-3) \end{align}

Note: If we simplify this point-slope formula, we obtain the slope-intercept form of the line: \begin{align} y-1 &=2(x-3) \\[3pt] y-1 &= 2x-6 \\[3pt] y &= 2x-5 \end{align}

Watch out! We could put (x_1,y_1)=(5,5). In this case, the point-slope form will be different: y-5=2(x-5) But the resulting simplified slope-intercept form will be the same: y = 2x-5

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Example: Finding a Point-Slope Equation Given a Slope and a Point

Find, in point-slope form, the equation of the line that passes through the point (6,2) and has a slope of 4.

EXPLANATION

Using m=4 and (x_1,y_1) = (6,2) , we substitute into the point-slope formula:

\begin{align} y - y_1 &= m(x-x_1)\\ y-2 &=4(x-6) \end{align}

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Practice: Finding a Point-Slope Equation Given a Slope and a Point

Find, in point-slope form, the equation of the line that passes through the point $(-3, 5)$ and has a slope of $7.$

a
 $y+3=7(x-5)$
b
 $y+5=7(x-3)$
c
 $y-5=7(x-3)$
d
 $y-5=7(x+3)$
e
 $y-3=7(x+5)$
Practice: Finding a Point-Slope Equation Given a Slope and a Point

Find, in point-slope form, the equation of the line that passes through the point $(2, 1)$ and has a slope of $3.$

a
 $y-2 = 3(x + 1)$
b
 $y-1 = 3(x - 2)$
c
 $y-1 = 3(x + 2)$
d
 $y-2 = 3(x - 1)$
e
 $y+1 = 3(x - 2)$
Example: Finding a Point-Slope Equation Given Two Points

Find, in point-slope form, the equation of the straight line that passes through the points (5,7) and (1,-1).

EXPLANATION

We know that the line passes through the point (5,7), so we can use (x_1,y_1) = (5,7) in our point-slope formula. All that remains is to compute m, and we can do this using the given points:

\begin{align} m = \frac{y_2 - y_1}{x_2 - x_1} =\frac{-1 - 7}{1 - 5} =\frac{-8}{-4} =2 \end{align}

Then, we substitute m=2 and the point (x_1,y_1) = (5,7) into the point-slope formula:

\begin{align} y - y_1 &= m(x-x_1)\\ y-7 &=2(x-5) \end{align}

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Practice: Finding a Point-Slope Equation Given Two Points

Find, in point-slope form, the equation of the line that passes through the points $(4,8)$ and $(2,12).$

a
 $y-8 =\dfrac 1 2(x-4)$
b
 $y-8 =-2(x-4)$
c
 $y+8 =\dfrac 1 2(x+4)$
d
 $y+8 =-2(x+4)$
e
 $y+8=-2(x-4)$
Practice: Finding a Point-Slope Equation Given Two Points

Find, in point-slope form, the equation of the line that passes through the points $(-2,1)$ and $(6,5).$

a
 $y-1=2(x-2)$
b
 $y-1=\dfrac{1}{2}(x-2)$
c
 $y-1=2(x+2)$
d
 $y+1=\dfrac{1}{2}(x-2)$
e
 $y-1=\dfrac{1}{2}(x+2)$
Example: Finding a Slope-Intercept Form Using Point-Slope Form

Calculate the equation of the straight line that passes through the points (1,10) and (3,-2), giving your final answer in slope-intercept form.

EXPLANATION

First, we will construct the point-slope formula for the line. Then, we will simplify the point-slope formula into slope-intercept form.

We start by computing the slope m using the given points:

\begin{align} m = \frac{y_2 - y_1}{x_2 - x_1} =\frac{-2 - 10}{3 - 1} =\frac{-12}{2} =-6 \end{align}

Then, we substitute m=-6 and the point (x_1,y_1) = (1,10) into the point-slope formula:

\begin{align} y - y_1 &= m(x-x_1)\\ y-10 &=-6(x-1) \end{align}

Finally, we simplify the equation into slope-intercept form:

\begin{align} y-10 &=-6(x-1) \\ y-10 &=-6x+6\\ y &=-6x+16 \end{align}

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Practice: Finding a Slope-Intercept Form Using Point-Slope Form

Calculate the equation of the line that passes through the points $(2,8)$ and $(3,12).$

a
 $y = 4x + 16$
b
 $y = 4x$
c
 $y = -4x +2$
d
 $y = 4x -8$
e
 $y = 4x +8$
Practice: Finding a Slope-Intercept Form Using Point-Slope Form

Calculate the equation of the line that passes through the points $(-4,2)$ and $(4,8).$

a
 $y=\dfrac{3}{4}x+2$
b
 $y=-\dfrac{3}{4}x+5$
c
 $y=\dfrac{3}{4}x+5$
d
 $y=\dfrac{3}{4}x+8$
e
 $y=-\dfrac{3}{4}x+8$
Practice: Finding a Slope-Intercept Form Using Point-Slope Form

Calculate the equation of the line that passes through the points $(4,2)$ and $(6,8).$

a
 $y=3x+2$
b
 $y=3x-10$
c
 $y=3x+10$
d
 $y=3x+8$
e
 $y=-3x+10$
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