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### 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:
$y-b=m\left(x-a\right)$
When an equation is written in this form, $m$ gives the slope of the line and $\left(a,b\right)$ 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 $\left(1,5\right)$ and whose slope is $-2$. Well, we simply plug $m=-2$, $a=1$, and $b=5$ into point-slope form!
$y-5=-2\left(x-1\right)$

### Example 2: Equation from two points

Suppose we want to find the line that passes through the points $\left(1,4\right)$ and $\left(6,19\right)$. First, we use the two points to find the slope:
Now we use one of the points, let's take $\left(1,4\right)$, and write the equation in point-slope:
$y-4=3\left(x-1\right)$
Problem 1
Write the point-slope equation of the line that passes through $\left(7,3\right)$ 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 $y-1=2\left(x-3\right)$. We can tell that the corresponding line passes through $\left(3,1\right)$ and has a slope of $2$. Now we can graph the line:
Problem 1
What is the slope of the line $y-5=-4\left(x-8\right)$?
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!
• how do you get that answer for number 2
• What do you mean for number 2? I guess I'll just explain both problems here.

Write the point-slope equation of the line that passes through (7,3) whose slope is 2.

This is pretty straightforward, since point-slope form requires you to just substitute values in order to form the equation.
Answer: y - 3 = 2(x - 7)

Write the point-slope equation of the line that passes through (3,5) and (7,1).

To solve this, you need to find the slope first. Slope for line connecting (x1, y1) and (x2, y2) is
(y1 - y2) / (x1 - x2).
Answer: y - 5 = -(x - 3) or y - 1 = -(x - 7)

What is the slope of the line y - 5 = -4(x - 8)?

Since both coefficient for both x and y are 1, you don't have to consider anything complicated.

To know which point does the line pass through, just substitute for x and y and compare L.H.S. with R.H.S.

Graph y - 7 = -3(x - 1)

There are many ways to do this, such as setting a point and doing some calculation but I will use another way. Recall slope is equal to change of y-coordinate for every one unit change in the x-coordinate. So if the slope is -3, it means for every 1 unit increase in x-coordinate, y-coordinate will reduce by 3 unit (-3). So we just take the point provided in the equation (1, 7) and take another point (1 + 1, 7 - 3) = (2, 4)
• So this is basically (change in y)=(slope)(change in x)?
• Yes, good observation.
• 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.
• Why do we learn about point-slope form in addition to the form y=mx+b? Seems like y=mx+b is better. What are the pros/cons of each form?
• With Point-slope form you can graph a line knowing any point and the slope. With slope-intercept form, you need the y-intercept(not just any point) to get the line. So if somebody tells you they can perform a job a a certain rate and gives you a certain moment(i.e. "after 25 minutes it will be 9 dollars") you immediately know the "graph" of the job. With slope-intercept, you would have to work to find the moment representing the y-intercept of the job's "graph".
Hopefully this helps. Tell me if my answer was not satisfactory.
• Im confused on how to find slope. any tips anyone?