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## Algebra 1

### Course: Algebra 1>Unit 5

Lesson 4: Point-slope form

# Point-slope form review

Review point-slope form and how to use it to solve problems.

## What is point-slope form?

Point-slope is a specific form of linear equations in two variables:
y, minus, start color #1fab54, b, end color #1fab54, equals, start color #ed5fa6, m, end color #ed5fa6, left parenthesis, x, minus, start color #11accd, a, end color #11accd, right parenthesis
When an equation is written in this form, start color #ed5fa6, m, end color #ed5fa6 gives the slope of the line and left parenthesis, start color #11accd, a, end color #11accd, comma, start color #1fab54, b, end color #1fab54, right parenthesis is a point the line passes through.
This form is derived from the slope formula.

## Finding point-slope equation from features or graph

### Example 1: Equation from slope and point

Suppose we want to find the equation of the line that passes through left parenthesis, start color #11accd, 1, end color #11accd, comma, start color #1fab54, 5, end color #1fab54, right parenthesis and whose slope is start color #ed5fa6, minus, 2, end color #ed5fa6. Well, we simply plug start color #ed5fa6, m, equals, minus, 2, end color #ed5fa6, start color #11accd, a, equals, 1, end color #11accd, and start color #1fab54, b, equals, 5, end color #1fab54 into point-slope form!
y, minus, start color #1fab54, 5, end color #1fab54, equals, start color #ed5fa6, minus, 2, end color #ed5fa6, left parenthesis, x, minus, start color #11accd, 1, end color #11accd, right parenthesis

### Example 2: Equation from two points

Suppose we want to find the line that passes through the points left parenthesis, 1, comma, 4, right parenthesis and left parenthesis, 6, comma, 19, right parenthesis. First, we use the two points to find the slope:
\begin{aligned} \text{Slope}&=\dfrac{19-4}{6-1} \\\\\ &=\dfrac{15}{5} \\\\ &=\maroonC 3 \end{aligned}
Now we use one of the points, let's take left parenthesis, start color #11accd, 1, end color #11accd, comma, start color #1fab54, 4, end color #1fab54, right parenthesis, and write the equation in point-slope:
y, minus, start color #1fab54, 4, end color #1fab54, equals, start color #ed5fa6, 3, end color #ed5fa6, left parenthesis, x, minus, start color #11accd, 1, end color #11accd, right parenthesis
Problem 1
• Current
Write the point-slope equation of the line that passes through left parenthesis, 7, comma, 3, right parenthesis whose slope is 2.

Want to try more problems like this? Check out this exercise.

## Finding features and graph from point-slope equation

When we have a linear equation in point-slope form, we can quickly find the slope of the corresponding line and a point it passes through. This also allows us to graph it.
Consider the equation y, minus, start color #1fab54, 1, end color #1fab54, equals, start color #ed5fa6, 2, end color #ed5fa6, left parenthesis, x, minus, start color #11accd, 3, end color #11accd, right parenthesis. We can tell that the corresponding line passes through left parenthesis, start color #11accd, 3, end color #11accd, comma, start color #1fab54, 1, end color #1fab54, right parenthesis and has a slope of start color #ed5fa6, 2, end color #ed5fa6. Now we can graph the line:
A first quadrant coordinate plane. The x- and y-axes each scale by one. The equation y minus one equals two times the difference of x minus three is graphed and labeled. The points three, one and four, three are plotted. There is a horizontal segment from three, one to four, one that is labeled plus one. There is a vertical segment from four, one to four, three that is labeled plus two.
Problem 1
• Current
What is the slope of the line y, minus, 5, equals, minus, 4, left parenthesis, x, minus, 8, right parenthesis?
The line passes through which point?

## Want to join the conversation?

• In school I learned it as y-y1=m(x-x1). Is it this just another way to write the same thing? Thank you
• Yes, it is essentially the same. Sal is just using the variables "a" and "b" instead of X1 and Y1.
• what is the slope of the line through (1,0) and (3,8)?
• Let's use the slope formula, where the slope (m) is equal to rise over run:

m = rise / run
= (y₂ - y₁) / (x₂ - x₁)
= (8 - 0) / (3 - 1)
= 8 / 2
= 4

m = 4, so the slope of the line through (1, 0) and (3, 8) is 4.

Hope this helps!
• I understand how this can create an equation, but not how it could be solved to an answer with two unknowns. If it can't be solved, what is the purpose exactly?
Thanks, Lilly
• What does it mean when the slope is simply - and you are trying to graph an equation?
• When the slope is . you should have to find it. You want to set up the problem like this:
(-1,3) (-2,5)
The 3 represents the y axis, and the -1 represents the x axis. you can interchange with the 2nd coordinate point. The 2 represents the slope. You can find the slope with slope-intercept form. Hope this helped.
• One line passes through the points \blueD{(-8,1)}(−8,1)start color #11accd, left parenthesis, minus, 8, comma, 1, right parenthesis, end color #11accd and \blueD{(4,4)}(4,4)start color #11accd, left parenthesis, 4, comma, 4, right parenthesis, end color #11accd. Another line passes through points \greenD{(-9,-7)}(−9,−7)start color #1fab54, left parenthesis, minus, 9, comma, minus, 7, right parenthesis, end color #1fab54 and \greenD{(9,-3)}(9,−3)start color #1fab54, left parenthesis, 9, comma, minus, 3, right parenthesis, end color #1fab54.
Are the lines parallel, perpendicular, or neither
• Im confused on how to find slope. any tips anyone?
• change in y / change in x
• So it says "now we use one of the y intercepts". So I chose one and I got it wrong because they chose the other one. Why did I get it wrong?
• Please help me understand what you are asking, In a linear equation, there is only one y intercept, so there is no choice as to which one you would choose, If you are just talking points, then anyone you choose should lead to the same answer, so I would have to see your process to find out where you went wrong, more than likely, it was a sign error somewhere.
I did not find anywhere where it talked about one of the y-intercepts, if you have more than one, then it is not a function and cannot be linear.
• How do i graph a straight line?? The KA graphing tool does not let me make a straight line... is this user error? Please help.
• If you are talking about the tool in the 2nd practice problem on this page, it only graphs a straight line. Drag the each dot to a point that you want to graph. Once you have each dot on the point you want, you have your line.
• If the point is (5,0) and the slope is 3, do you think we should use y-0=3(x-5) or just y=3(x-5)?
• Strict point slope form is: y-0=3(x-5)
It becomes y=3(x-5) if you simplify the left side.
• Hey! Just a random question,
if you get result of point-slope form
y-9=-4(x-4)
And you want to convert to slope-intercept form it will be:
y = -4x+25
, right?
But if i want to convert:
y = -4x+25
to point-slope form again, it can be
*y-25 = -4(x-0)* right?
If so is this right or wrong? or is there some kind of method to turn slope-intercept form to point-slope form?