Looking to understand the three forms of linear equations? In this post, we learn about the three forms of linear equations and how to convert from one form to the next.

Mastering new math skills is a lot like adding a new tool to your toolbelt. Yes, it may take time to understand how the new tool works, but new tools allow you to complete more intricate and exciting projects!

Learning how to use all three forms of linear equations might feel like learning how to use a swiss army knife. There are different functions and each one has a different purpose. It is important to use the right tool at the right time.

Let’s dive in!

What are the three forms of linear equations?

Linear equations can be written in three basic forms:

1. Slope-intercept form
2. Point-slope form
3. Standard form

Slope-intercept form easily identifies the slope and y-intercept of a line. For more info, read our full review on the slope-intercept form.

Slope-Intercept Form y=mx+b

Point-slope form is determined by one point and the slope of the line. For more info, read our full review on point-slope form.

Point-Slope Form y-y_1=m(x-x_1)

Standard form is useful for solving systems of equations and determining both x and y-intercepts.  For more info, read our full review on standard form.

Standard Form ax+by=c

A few notes on Standard Form:

• The a term must be a positive integer
• a, b, and c cannot be decimals or fractions

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Example: forms of linear equations from graph

All three forms of linear equations can describe the graph of a line. Let’s determine the linear equation of the following graph:

Slope-intercept form

To determine slope-intercept form, y=mx+b, we must input the slope and the y-intercept.

• The slope of the line is 3 because the graph goes up 3 units for every unit it goes to the right.
• The y-intercept is -1 because the line crosses the y-axis when y=-1.

Therefore, the equation of the line in slope-intercept form is:

y=3x-1

Point-slope form

To find point-slope form, y-y_1=m(x-x_1), we must input the slope and one point on the graph.

• We already know the slope of the line is 3 (see above).
• We can see that the line goes through the point (1,2). We will substitute 2 for y_1 and 1 for x_1.

Therefore, the equation of the line in point-slope form is:

y-2=3(x-1)

Standard form

To calculate standard form, ax+by=c, we will simply convert it from slope-intercept form to standard form.

y=3x-1

y-3x=-1

Therefore, the equation of the line in standard form is:

y-3x=-1

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What is a linear equation?

A linear equation is an equation describing a straight line.

The line can be defined by a point on the line and the slope or by any two points on the line. A linear equation cannot be used to describe a line where the slope changes or any graph that is curved.

For more details on what makes an equation “linear”, read this helpful article.

Convert point-slope to slope-intercept

In order to convert to slope-intercept form from point-slope form, we need to reorganize the equation. We can begin with the equation, y+4=2(x-13), which is in point-slope form.

First, we will distribute. Then, we will isolate the variable y.

y+4=2(x-13)

y+4=2x-26

y=2x-30

The equation y+4=2(x-13) is the slope-intercept form of the equation y+4=2(x-13).

For more visual learners, here’s a quick video example:

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Convert point-slope to standard form

Let’s keep exploring the forms of linear equations with another example!

We’re now going to convert the equation y+4=\frac{1}{2}(x-13), which is in point-slope form, to standard form.

We will again begin by distributing and isolating the variable y. Then, we will move the term with the variable x to the other side. Lastly, we will eliminate any rational numbers by multiplying by the denominator.

y+4=\dfrac{1}{2}(x-13)

y+4=\dfrac{x}{2}-\dfrac{13}{2}

y=\dfrac{x}{2}-\dfrac{13}{2}-4

y=\dfrac{x}{2}-\dfrac{5}{2}

y-\dfrac{x}{2}=-\dfrac{5}{2}

2y-x=-5

The equation 2y-x=-5 is the standard form of the equation y+4=\frac{1}{2}(x-13).

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Convert slope-intercept to point-slope

Let’s now begin in slope-intercept form, using the equation y=14x-5. To convert to point-slope form, we need to factor out the constant in front of x.

y=14x-5

y=14(x-\frac{5}{14})

It’s now in point-slope form, using the point (\frac{5}{14},0). To see it more clearly, you can write it as y-0=14(x-\frac{5}{14}). Therefore, y-0=14(x-\frac{5}{14}) is the point-slope form of the equation y=14x-5.

Keep in mind, there is more than one correct solution when converting to point slope form, as a line has an infinite number of points.

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Convert slope-intercept to standard form

Converting from slope-intercept to standard form requires very few steps. Let’s convert the slope-intercept equation y=-\frac{2}{3}x+14 into standard form.

We will simply move the term with x to the side with y and multiply by the denominator to eliminate rational numbers.

y=-\dfrac{2}{3}x+14 

y+\dfrac{2}{3}x=14

3y+2x=42

The equation 3y+2x=42 is the standard form of the equation y=-\frac{2}{3}x+14.

Here’s a quick video example of converting slope-intercept to standard form:

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Convert standard form to point-slope

Next up, we can convert from standard form to point-slope form. Let’s convert the equation 6x+2y=11 into point-slope form.

We will begin by moving the term with x to the other side of the equation. Then, we will factor out the coefficient in front of x, and finally, we will divide by the coefficient in front of y.

6x+2y=11

2y=-6x+11

2y=-6(x-\frac{11}{6})

y=-3(x-\frac{11}{6})

As shown previously, we can also write this equation as y-0=-3(x-\frac{11}{6}) to more clearly see the point used is (\frac{11}{6},0). Just a reminder, there is more than one correct solution when converting to point-slope form, as a line has an infinite number of points.

The equation y-0=-3(x-\frac{11}{6}) is a point-slope form version of the equation 6x+2y=11.

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Convert standard form to slope-intercept

Next, we can also change standard form to slope-intercept form. Let us change the equation 2x+5y=10 into slope-intercept form.

To do so, we must solve the equation for y. To solve for y, we move the term with x to the other side and divide by the coefficient in front of y.

2x+5y=10 

5y=-2x+10 

y=-\dfrac{2}{5}x+2 

The equation y=-\frac{2}{5}x+2 is the slope-intercept form of the equation 2x+5y=10.

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Summary: Forms of Linear Equations

• There are three common forms of linear equations
• The same graph of a line can be written in different ways using different forms of equations
• We can use algebraic skills to convert among the different forms of equations

Click here to explore more helpful Albert Algebra 1 review guides.