Finding the Point-Slope Form Equation

Hello! Today we’re going to take a look at how to write an equation for a line in point-slope form when we are given a point on a line and its slope, and when we are given two points on the line. Remember, point-slope form is:

\(y-y_{1}=m(x-x_{1})\)

where \((x_{1},y_{1})\) is a point on the line, and \(m\) is the slope of the line.

So, let’s start by finding the equation of the line that passes through the point \((2,4)\) and has a slope of 7.

Point: \((2,4)\) Slope: \(7\)

When we are given these two things, all we have to do is plug in the values where they belong. First, let’s write out our formula again:

\(y-y_{1}=m(x-x_{1})\)

Then we’re going to write out the values that we’re given:

\(x_{1}=2\)
 
\(y_{1}=4\)
 
\(m=7\)

So, if we plug in our values into this equation, we’ll get:

\(y-4=7(x-2)\)

So, our equation for this line is \(y-4=7(x-2)\). Let’s try another one!

Find the point-slope form equation for a line that passes through the point \((-6,3)\) and has a slope of 2.

Point: \((-6,3)\) Slope: \(2\)

First, write out the values you are given.

\(x_{1}=-6\)
 
\(y_{1}=3\)
 
\(m=2\)

Then we’re going to write out our formula, this will help us remember and make sure that we plug our values into the correct place.

\(y-y_{1}=m(x-x_{1})\)

Now, plug in the values you are given.

\(y-3=2(x-(-6))\)

Finally, simplify the subtracting a negative part. Remember, when you subtract a negative number, it’s the same thing as adding that number, so we have:

\(y-3=2(x+6)\)

And it’s as simple as that!

Now we’re going to take a look at how we find the point-slope form equation when we’re given two points on the line.

Points: \((8,5)\) and \((-4,11)\)

Remember, when we’re given points on a line, that means that the line passes through these points. So, to write an equation in point-slope form, we need to know a point on the line and the slope of the line. We’re given two points to work with, but we don’t know the slope. To find the slope, we need to use the slope formula.

\(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

For these points, let’s call \((8,5)\) the point \((x_{1},y_{1})\) and \((-4,11)\) the point \((x_{2},y_{2})\). If you wanted to switch the points and have \((-4,11)\) be \((x_{1},y_{1})\) and \((8,5)\) be \((x_{2},y_{2})\), that’s also perfectly acceptable. You’ll get the right answer either way.

Now, let’s plug our values into the slope formula and solve for \(m\).

\(m=\frac{11-5}{-4-8}=\frac{6}{-12}=-\frac{1}{2}\)

Now that we know our slope, let’s choose one of the two points to write our formula. For this example, I’m going to use the point \((8,5)\). You can absolutely use \((-4,11)\) if you want to, and you will get a correct answer as well. It won’t be the exact same answer as using the point \((8,5)\), but if you follow the exact same steps, it will also be correct.

Let’s first list the values we are given.

\(x_{1}=8\)
 
\(y_{1}=5\)
 
\(m=-\frac{1}{2}\)

Now, let’s write our formula out.

\(y-y_{1}=m(x-x_{1})\)

And now we can plug in our values.

\(y-5=-\frac{1}{2}(x-8)\)

There’s our answer! If you used the point \((-4,11)\) instead, your answer should be \(y-11=-\frac{1}{2}(x+4)\). Either one of these two equations is correct.

Let’s try one more together before we go. Find the point-slope form equation for the line that passes through the points \((6,-1)\) and \((8,7)\).

First, we need to find the slope of the line. Let’s say the point \((6,-1)\) is \((x_{1},y_{1})\) and the point \((8,7)\) is \((x_{2},y_{2})\). First, write out the slope equation.

\(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Now, plug in the values.

\(m=\frac{7-(-1)}{8-6}=\frac{7+1}{8-6}=\frac{8}{2}=\frac{4}{1}=4\)

Now that we know the slope is 4, choose either one of the points to plug into the formula. For this example, let’s use the point \((6,-1)\). First, we’re going to write out our known values.

\(x_{1}=6\)
 
\(y_{1}=-1\)
 
\(m=4\)

Then we’re going to write out our formula.

\(y-y_{1}=m(x-x_{1})\)

Then we’re going to plug in our known values.

\(y-(-1)=4(x-6)\)

Finally, we’re going to simplify the left side, so we no longer have a subtracting a negative number. So, we’re going to make this:

\(y+1=4(x-6)\)

If you chose the point \((8,7)\) instead, your answer should be \(y-7=4(x-8)\). Again, either one of these two formulas is an acceptable answer. So, \(y-7=4(x-8)\) and \(y+1=4(x-6)\) are both correct answers.

I hope this video helped you get more comfortable with point-slope form. Thanks for watching, and happy studying!