Equation of a Line Given Point and Slope

Learning Objectives

Write and solve equations of lines using slope and a point on the line
- Write the equation of a line given the slope and a point on the line.
- Identify which parts of a linear equation are given and which parts need to be solved for using algebra

Find the Equation of a Line Given the Slope and a Point on the Line

Using the slope-intercept equation of a line is possible when you know both the slope (m) and the y-intercept (b), but what if you know the slope and just any point on the line, not specifically the y-intercept? Can you still write the equation? The answer is yes, but you will need to put in a little more thought and work than you did previously.

Recall that a point is an (x, y) coordinate pair and that all points on the line will satisfy the linear equation. So, if you have a point on the line, it must be a solution to the equation. Although you don’t know the exact equation yet, you know that you can express the line in slope-intercept form, $y=mx+b$ .

You do know the slope (m), but you just don’t know the value of the y-intercept (b). Since point (x, y) is a solution to the equation, you can substitute its coordinates for x and y in $y=mx+b$ and solve to find b!

This may seem a bit confusing with all the variables, but an example with an actual slope and a point will help to clarify.

Example

Write the equation of the line that has a slope of 3 and contains the point $(1,4)$ .

Show Solution

Substitute the slope (m) into $y=mx+b$ .

$y=3x+b$

Substitute the point $(1,4)$ for x and y.

$4=3\left(1\right)+b$

Solve for b.

$\begin{array}{l}4=3+b\\1=b\end{array}$

Rewrite $y=mx+b$ with $m=3$ and $b=1$ .

Answer

$y=3x+1$

To confirm our algebra, you can check by graphing the equation $y=3x+1$ . The equation checks because when graphed it passes through the point $(1,4)$ .

An uphill line passes through the y-intercept of (0,1) and the point (1,4). The rise is 3 and the run is 1.

If you know the slope of a line and a point on the line, you can draw a graph. Using an equation in the point-slope form allows you to identify the slope and a point. Consider the equation $\displaystyle y=-3x-1$ . The y-intercept is the point on the line where it passes through the y-axis. What is the value of x at this point?

Reminder: All y-intercepts are points in the form

(0, y)

$(0,y)$ . The x value of any y-intercept is always zero.

Therefore, you can tell from this equation that the y-intercept is at $(0,−1)$ , check this by replacing x with 0 and solving for y. To graph the line, start by plotting that point, $(0,−1)$ , on a graph.

You can also tell from the equation that the slope of this line is $−3$ . So start at $(0,−1)$ and count up 3 and over $−1$ (1 unit in the negative direction, left) and plot a second point. (You could also have gone down 3 and over 1.) Then draw a line through both points, and there it is, the graph of $\displaystyle y=-3x-1$ .

A downhill line passes through the point (-1,2) and the y-intercept (0,-1). The rise is 3 and the run is -1.

Example (Advanced)

Write the equation of the line that has a slope of $\frac{7}{8}$ and contains the point $\left(4,\frac{5}{4}\right)$ .

Show Solution

Substitute the slope (m) into $y=mx+b$ .

$\begin{array}{l}y=mx+b\\\\y=-\frac{7}{8}x+b\end{array}$

Substitute the point $\left(4,\frac{5}{4}\right)$ for x and y.

$\frac{5}{4}=-\frac{7}{8}\left(4\right)+b$

Solve for b.

$\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\frac{5}{4}=-\frac{28}{8}+b\\\\\,\,\,\,\,\,\,\,\,\,\,\,\frac{5}{4}=-\frac{14}{4}+b\\\\\frac{5}{4}+\frac{14}{4}=-\frac{14}{4}+\frac{14}{4}+b\\\\\,\,\,\,\,\,\,\,\,\,\frac{19}{4}=b\end{array}$

Rewrite $y=mx+b$ with $\displaystyle m=-\frac{7}{8}$ and $\displaystyle b=\frac{19}{4}$ .

Answer

$y=-\frac{7}{8}x+\frac{19}{4}$

In this video, you will see an additional example of how to find the equation of a line given the slope and a point.

Objective 4

Learning Objectives

Find the Equation of a Line Given the Slope and a Point on the Line

Example

Answer

Example (Advanced)

Answer

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