Writing Linear Equations in Point-Slope Form

The equation of a line can be written in slope-intercept form or standard form. However, when the slope and a point are given, point-slope form is another alternative.

$y-y_1 = m(x-x_1)$

Here,

m

is the slope and

(x_1,y_1)

is a point on the line. By simplifying and solving for

y,

it's possible to rewrite a point-slope equation in slope-intercept form:

y = mx + b.

Method

Writing a Line in Point-Slope Form

As its name suggests, point-slope form requires a point on and the slope of the line.

To write the point-slope form for the line that passes through the points

(\text{-}1,5) \quad \text{and} \quad (1,1),

find the slope, then use either point to write the equation.

1

Find the slope

First, it can be helpful to determine the slope of the line,

m.

Depending on what information about the line is given, finding the slope could be done both algebraically and graphically. In this case, two points are given, so the slope formula is preferable.

m = \dfrac{y_2-y_1}{x_2-x_1}

SubstitutePointsSubstitute

\left({\color{#0000FF}{1,1}}\right)

&

\left({\color{#009600}{\text{-}1,5}}\right)

m = \dfrac{{\color{#0000FF}{1}}-{\color{#009600}{5}}}{{\color{#0000FF}{1}}-\left( {\color{#009600}{\text{-}1}} \right)}

SubTermSubtract term

m=\dfrac{\text{-}4}{1-(\text{-}1)}

SubNeg

a-(\text{-} b)=a+b

m=\dfrac{\text{-}4}{2}

CalcQuotCalculate quotient

m=\text{-}2

The slope of the line that passes through the given points is

m=\text{-} 2.

2

Choose one point on the line

The next step is to choose one point on the line to use. If the line is given as a graph, find a point whose coordinates are easy to identify. In this case, two points are given. Either can be used. For simplicity, here, use $(1,1).$

3

Substitute values

Now that the slope has been found and a point has been chosen, the equation of the line can be written by substituting the corresponding values. Here, substitute $m={\color{#0000FF}{\text{-} 2}}$ and $(x_1,y_1)=({\color{#009600}{1}},{\color{#009600}{1}}).$ $y-y_1=m(x-x_1) \quad \Rightarrow \quad y-{\color{#009600}{1}}={\color{#0000FF}{\text{-}2}}(x-{\color{#009600}{1}}).$

Write the linear equation in point-slope form using the graph

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Exercise

Write the equation for the line in point-slope form.

Show Solution

Solution

To write the equation of the line in point-slope form, $y-y_1=m(x-x_1),$ we need the slope and a point.

Example

The slope

Since we're given the line as a graph we can use the rise and run to find the slope. We'll find the rise and run between two arbitratily chosen points, the $y$ -intercept, $(0,\text{-} 2),$ and $(2,1).$

Here, the run is $2$ and the rise is $3,$ which gives a slope of $m=\dfrac{3}{2}.$

Example

Choosing a point

Next, any point on the line can be used for $(x_1,y_1).$ Let's use the same point as above, $(2,1).$

Example

Writing the equation

Lastly, substitute the found values of $m$ and $(x_1,y_1)$ into $y-y_1=m(x-x_1).$ Here, $m={\color{#0000FF}{\frac{3}{2}}}$ and $(x_1,y_1)=({\color{#009600}{2}},{\color{#009600}{1}})$ will be substituted. $y-{\color{#009600}{1}}={\color{#0000FF}{\dfrac{3}{2}}}(x-{\color{#009600}{2}}).$

Write the linear equation in point-slope form using the table

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Exercise

All points in the table lie on the same line.

Write the equation of the line in point-slope form.

Show Solution

Solution

To write an equation in the form $y-y_1=m(x-x_1),$ the slope of the line, $m,$ and any point on the line, $(x_1,y_1),$ must be known. We'll begin by finding the slope between consecutive points. It can be seen that the difference between each $x$ -value is $1.$