4.6 Find the equation of a line  (Page 2/7)

We are going to use the slope formula to derive another form of an equation of the line. Suppose we have a line that has slope $m$ and that contains some specific point $\left(x_1,y_1\right)$ and some other point, which we will just call $\left(x,y\right)$ . We can write the slope of this line and then change it to a different form.

$\begin{array}{cccccc}& & & \hfill m& =\hfill & \frac{y-y_1}{x-x_1}\hfill \\ \text{Multiply both sides of the equation by}x-x_1.\hfill & & & \hfill m\left(x-x_1\right)& =\hfill & \left(\frac{y-y_1}{x-x_1}\right)\left(x-x_1\right)\hfill \\ \text{Simplify.}\hfill & & & \hfill m\left(x-x_1\right)& =\hfill & y-y_1\hfill \\ \text{Rewrite the equation with the}y\text{terms on the left.}\hfill & & & \hfill y-y_1& =\hfill & m\left(x-x_1\right)\hfill \end{array}$

This format is called the point–slope form of an equation of a line.

Point–slope form of an equation of a line

The point–slope form    of an equation of a line with slope $m$ and containing the point $\left(x_1,y_1\right)$ is

We can use the point–slope form of an equation to find an equation of a line when we are given the slope and one point. Then we will rewrite the equation in slope–intercept form. Most applications of linear equations use the the slope–intercept form.

Find an equation of a line given the slope and a point

Find an equation of a line with slope $m=\frac{2}{5}$ that contains the point $\left(10,3\right)$ . Write the equation in slope–intercept form.

Solution

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Find an equation of a line with slope $m=-\frac{1}{3}$ that contains the point $\left(6,\mathrm{-4}\right)$ . Write the equation in slope–intercept form.

Solution

Since we are given a point and the slope of the line, we can substitute the needed values into the point–slope form, $y-y_1=m\left(x-x_1\right)$ .

Identify the slope.
Identify the point.
Substitute the values into $y-y_1=m(x-x_1).$
Simplify.
Write in slope–intercept form.

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Find an equation of a horizontal line that contains the point $\left(\mathrm{-1},2\right)$ . Write the equation in slope–intercept form.

Solution

Every horizontal line has slope 0. We can substitute the slope and points into the point–slope form, $y-y_1=m\left(x-x_1\right)$ .

Identify the slope.
Identify the point.
Substitute the values into $y-y_1=m(x-x_1).$
Simplify.
Write in slope–intercept form.	It is in y -form, but could be written $y=0x+2$ .

Did we end up with the form of a horizontal line, $y=a$ ?

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Find an equation of the line given two points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.

We have two options so far for finding an equation of a line: slope–intercept or point–slope. Since we will know two points, it will make more sense to use the point–slope form.

But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.