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, [latex]y=mx+b[/latex].

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 [latex]y=mx+b[/latex] 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 [latex](1,4)[/latex].

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

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 [latex]\displaystyle y=-3x-1[/latex]. 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 [latex](0,y)[/latex].  The x value of any y-intercept is always zero.

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

You can also tell from the equation that the slope of this line is [latex]−3[/latex]. So start at [latex](0,−1)[/latex] and count up 3 and over [latex]−1[/latex] (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 [latex]\displaystyle y=-3x-1[/latex].

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 [latex]\frac{7}{8}[/latex] and contains the point [latex]\left(4,\frac{5}{4}\right)[/latex].

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