Write the equation of the line that passes through $\hspace{0.2em} (-1, 4) \hspace{0.2em}$ and has a slope of $\hspace{0.2em} -3 \hspace{0.2em}$.

In this tutorial, we’ll focus on the equation of a line in the point-slope form. Let’s dive in.

The point-slope form is one of the different forms in which an equation of a line can be written.

This form is most useful when you want to write the equation of a line when its slope and the coordinates of a point on the line are known.

If a line passes through a given point $\hspace{0.2em} (x_1, y_1) \hspace{0.2em}$ and has a slope m, its equation in the point-slope form is given by –

Here’s a visual representation.

Consider the figure below.

We have a line with a slope $\hspace{0.2em} (m) \hspace{0.2em}$ and the line passes through a point with coordinates $\hspace{0.2em} (x_1, y_1) \hspace{0.2em}$. Also $\hspace{0.2em} (x, y) \hspace{0.2em}$ are the coordinates of any arbitrary point on the line.

Let’s get the expression for the slope $\hspace{0.2em} m \hspace{0.2em}$ of the line. We know that

$\text{slope} = \frac{\text{rise}}{\text{run}}$

Substituting the values from the figure

$m = \frac{y - {\color{Red} y_1} }{x - {\color{Red} x_1} }$

Multiplying both sides by $\hspace{0.2em} x - x_1 \hspace{0.2em}$, we have

$y - {\color{Red} y_1} = m(x - {\color{Red} x_1} )$

And just like that, we have derived the point-slope form equation of a line.

Example

Write the equation of the line that passes through $\hspace{0.2em} (-1, 4) \hspace{0.2em}$ and has a slope of $\hspace{0.2em} -3 \hspace{0.2em}$.

Solution

We know the equation of a line in its point-slope form is

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

And the question tells us –

$\begin{align*} m &= -3 \\[1em] (x_1, y_1) &= (-1, 4) \end{align*}$

So to get the equation of our line, we just need to substitute the values in our standard equation.

$\begin{align*} y - 4 &= -3(x - (-1)) \\[1em] y - 4 &= -3(x + 1) \end{align*}$

Voila! That’s the equation we wanted.

Example

Write the equation of the line shown in the figure.

Write the equation of the line shown in the figure.

Solution

This example is not very different from the previous one – except that this time, we need to extract info from the figure.

It tells us the slope of the line is

$m = -\frac{2}{3}$

The point marked in the figure is $\hspace{0.2em} (5, 0) \hspace{0.2em}$. How?

It’s on the x-axis, so the y-coordinate is 0. And it’s at a distance of 5 units from the y-axis, so the x-coordinate is 5.

That means –

$(x_1, y_1) = (5, 0)$

So again, substituting the above values in our standard equation, we get

$\begin{align*} y - y_1 &= m(x - x_1) \\[1.3em] y - 0 &= -\frac{2}{3}(x - 5) \\[1.3em] y &= -\frac{2}{3}(x - 5) \end{align*}$

Now it’s time to see how the point-slope form can be used to derive the slope-intercept form.

Consider the figure below.

We have a line with a slope $\hspace{0.2em} m \hspace{0.2em}$ and a $\hspace{0.2em} y \hspace{0.2em}$-intercept of $\hspace{0.2em} b \hspace{0.2em}$ (meaning it cuts the $\hspace{0.2em} y \hspace{0.2em}$-axis at a distance of $\hspace{0.2em} b \hspace{0.2em}$ units from the origin).

Let’s get the equation of the line using the point-slope form.

Now, we can do something very similar to what we did for the previous example and get the equation of the line above.

We start with the equation in the point-slope form and proceed from there.

$\begin{align*} y - y_1 &= m(x - x_1) \\[1em] y - {\color{Red} b} &= {\color{Teal} m} (x - {\color{Red} 0} ) \\[1em] y &= {\color{Teal} m} x + {\color{Red} b} ) \end{align*}$

Looks familiar, doesn’t it? It’s the equation of a line in the slope-intercept form.

And with that, we come to the end of this tutorial on the point-slope form equation of a line.

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