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### Course: Algebra (all content)>Unit 3

Lesson 10: Summary: Forms of two-variable linear equations

# Forms of linear equations review

There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form. We review all three in this article.
There are three main forms of linear equations.
Slope-interceptPoint-slopeStandard
$y=mx+b$$y-{y}_{1}=m\left(x-{x}_{1}\right)$$Ax+By=C$
where $m$ is slope and $b$ is the $y$-interceptwhere $m$ is slope and $\left({x}_{1},{y}_{1}\right)$ is a point on the linewhere $A$, $B$, and $C$ are constants

## Example

A line passes through the points $\left(-2,-4\right)$ and $\left(-5,5\right)$. Find the equation of the line in all three forms listed above.
Two of the forms require slope, so let's find that first.
$\begin{array}{rl}\text{slope}=m& =\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\\ \\ & =\frac{5-\left(-4\right)}{-5-\left(-2\right)}\\ \\ & =\frac{9}{-3}\\ \\ & =-3\end{array}$
Now we can plug in $m$ and one of the points, say $\left(-5,5\right)$, to get point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$:
$\begin{array}{rl}y-{y}_{1}& =m\left(x-{x}_{1}\right)\\ \\ y-5& =-3\left(x-\left(-5\right)\right)\\ \\ y-5& =-3\left(x+5\right)\end{array}$
Solving for $y$, we get slope-intercept form, $y=mx+b$:
$\begin{array}{rl}y-5& =-3\left(x+5\right)\\ \\ y-5& =-3x-15\\ \\ y& =-3x-10\end{array}$
And adding $3x$ to both sides, we get standard form, $Ax+By=C$:
$y+3x=-10$
Want another example? Check out this video.
Want to practice the different forms yourself? Check out this exercise.
Want a more in-depth review of each form? Check out these review articles:

## Want to join the conversation?

• In the point slopes form, it looks like you're saying you could use either set of coordinates.I thought it was the first set of coordinates since it says x1 and y1. Please explain. Thanks.
• That is correct. You can definitely use either set of coordinates. Don't mix-and-match: you can't use x1 and y2, but you can use (x1, y1) or (x2, y2) and it will work just as well either way.
• How do you know when to use point slope form vs slope intercept form?
• Most of the time, it would be your choice. Though, your teacher may request that you use a specific approach to see if you know how to do it.
• ax + by + c = 0
ax + by = c

I've heard of 2 "standard" forms of linear equations. Which one is correct?

should c in the 1st line be -c though? since im moving it from the right to left...?
• hey! okay, so I'm pretty sure you're confusing a quadratic equation with a linear equation. A linear equation is a straight line, while a quadratic is a curve/parabola. You'll probably learn that later in algebra 1 and 2.

anyways, the standard linear equation is ax+by=c, while the standard quadratic equation is slightly different from what you have; it should be ax^2+bx+c=0

hope this helps!!
• when do you need to use slope?
• To determining the slope/ steepness of a line. You should review the slope videos if you need help.
(1 vote)
• i must be behind in math because all of this is way too confusing
• In the previous exercise: "Linear equations in any form", is there a method to figure out from the graph the equation in standard form directly or do you have to work out one of the slope forms first and then re-arrange the formula?
• You would need to use point-slope form or slope intercept form to create an equation. Then, convert it to standard form.
• Why are point-slope operations the opposite?
For example, the point is (2,-3).
why is y+3=3/4(x-2) correct but not y-3=3/4(x+2)?
• Point slope form is a variation of the slope formula:
Slope m = (y2-y1)/(x2-x1)
If you mulitply both sides by (x2-x1), then you get point slope form:
(y2-y1) = m(x2-x1)
Then, they swab a couple of variables to clarify the variables that stay. X2 becomes X, and Y2 becomes Y. And, you have the point slope form.

Remember, slope is calculated as the change in Y over the change in X. So, it requires the subtraction.

Hope this helps.