Point-Slope Form (of a Linear Equation)

You've previously learned that any linear function can be graphed and any line has at least one corresponding function, but looking at a line often isn't enough to determine what the equation is. Fortunately, you can determine the equation of any line if you know its slope (m) and the coordinates of one point $\left(x_1,y_1\right)$ . This also allows the equation to be easily written in the point-slope form of a line.

The point-slope form formula

The formula for the point-slope form is as follows:

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

That looks pretty abstract, so it might help if we plug actual numbers into the equation to see how it works. For example, let's try to find the equation of a line with slope $-\frac{1}{2}$ passing through the point $\left(-3,2\right)$ .

The slope is $-\frac{1}{2}$ , so we can plug that in for m. Likewise, -3 is our x₁ and 2 is our y₁. That gives us the following equation:

$y-2=-\frac{1}{2}(x-\left(-3\right))$

While correct, there is still some simplifying to do to get it into its simplest form. Therefore, we want to combine the two minus signs into a plus sign to get a mathematically equivalent expression that looks a bit cleaner:

$y-2=-\frac{1}{2}(x+3)$

Here is an example of what this line looks like once graphed:

Always remember to extend your lines in both directions to show that it's longer than what you've drawn!

The relationship between point-slope form and slope-intercept form

If the point you're using to find a line's point-slope form is its y-intercept, it's a special case called the slope-intercept form. The standard format of the slope-intercept form is $y=mx+b$ where m is the slope and b is the y-intercept. If you have a line with a slope of 2 and a y-intercept of $\left(0,3\right)$ , you can write its equation in slope-intercept form as:

$y=2x+3$

This format makes it easier to graph, so feel free to use it whenever the point you have is the y-intercept $\left(0,b\right)$ .

Point-slope form practice questions

a. Write an equation in point-slope form for a line with a slope of 3 that passes through $\left(5,4\right)$

$y-4=3(x-5)$

b. Write an equation in point-slope form for a line with a slope of -2 that passes through $\left(-2,-3\right)$ .

$y-\left(-3\right)=-2(x-\left(-2\right))$

$y+3=-2(x+2)$

c. Write an equation in point-slope form for a line with a slope of 5 that passes through $\left(0,3\right)$ . Hint: the point lies on the y-axis.

$y-3=5(x-0)$

Or using the y intercept trick to jump to slope intercept form:

$y=5x+3$

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