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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 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 x_{1}
and 2 is our y_{1}. 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!

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 $\left(0,b\right)$ .

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:

$y=5x+3$

Writing Systems of Linear Equations from Word Problems

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Many students have a hard time with abstract math concepts, and point-slope form isn't a very intuitive topic. If your student is falling behind their peers or you just want to give them access to all of the educational tools you can, 1-on-1 tutoring provides a safe environment where students can ask questions and express interest in mathematics while learning at their pace. Whether your student needs extra practice questions or a different voice to explain how math works, the Educational Directors at Varsity Tutors can explain the many benefits of private instruction and find a qualified tutor to work with your student. Get in touch today.

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