Standard form of linear equations

A linear equation is an equation whose graph is a straight line (not a curve). In this chapter, you’ll work with three common forms of linear equations:

• standard form
• point-slope form
• slope-intercept form

A quick way to recognize a linear equation is to check the exponents on the variables: an equation is linear if the highest exponent on any variable is $1$.

Standard form

The standard form of a linear equation is:

$$Ax+By=C$$

Here, $x$ and $y$ are variables, and $A$, $B$, and $C$ are integers.

This form is useful because it makes intercepts easy to find:

• To find the $x$-intercept, set $y=0$ and solve for $x$.
• To find the $y$-intercept, set $x=0$ and solve for $y$.

A helpful way to remember this: solve for the same variable as the intercept you want. For the $y$-intercept, you’ll end up solving for $y$.

You can also find the slope directly from standard form. If the equation is $Ax+By=C$, then the slope is

$$m=-\frac{A}{B}$$

And if you want the $y$-intercept without solving for $y$, you can use

$$\frac{C}{B}$$

We’ll work through a few examples to practice recognizing linear equations and finding intercepts.

Is the following equation a linear equation? If so, find the $x$ and $y$ intercepts.

$$4x-2y=9$$

Is the highest exponent on the variables $1$? Yes.

This is a linear equation. The minus sign doesn’t change that.

$x$-intercept: set $y=0$

$$4x-2(0)=9$$

Solve for $x$:

$$x=9/4$$

The $x$-intercept is $9/4$.

$y$-intercept: set $x=0$

$$4(0)-2y=9$$

Solve for $y$:

$$y=-9/2$$

The $y$-intercept is $-9/2$.

Try a few more examples before moving on. Notice that an equation can still be linear even if it isn’t written in standard form, as long as you can rearrange it into standard form.

Is the following equation a linear equation? If so, find the $x$ and $y$ intercepts.

$$y+3=3x$$

Is the following equation a linear equation? If so, find the $x$ and $y$ intercepts.

$$4-x^2=y$$

Point-slope form

The second form of linear equations is point-slope form. This form is useful when you want to find the slope between two points. You might be given two points directly, see two points on a graph, or find two points from an equation.

A linear equation in point-slope form looks like:

$$(y_2-y_1)=m(x_2-x_1)$$

Here:

• $m$ is the slope
• $(x_1,y_1)$ is the first point
• $(x_2,y_2)$ is the second point

This comes from the slope formula:

$$\text{slope}=\frac{(y_2-y_1)}{(x_2-x_1)}$$

A common way to remember slope is “rise over run”:

• $(y_2-y_1)$ is the rise (change in $y$)
• $(x_2-x_1)$ is the run (change in $x$)

Try a few examples.

What is the slope of the line formed by the points $(1,2)$ and $(3,5)$?

Rise over run!

How far does the line “rise” from the first point to the second point?

$5-2=3=\text{rise}$.

How far does the line “run” from the first point to the second point?

$3-1=2=\text{run}$.

Slope is rise over run:

$$m=\frac{3}{2}$$

Notice that “rise over run” is the same calculation as the slope formula. Either method works.

Find the slope of the following line:

Slope-intercept form

The third form is slope-intercept form. It’s useful because you can identify the slope and the $y$-intercept just by looking at the equation - as long as it’s written in the correct form.

Slope-intercept form is:

$$y=mx+b$$

where:

• $m$ is the slope
• $b$ is the $y$-intercept

The letters $m$ and $b$ are used here (instead of $A$ and $B$) to emphasize that the equation must be in slope-intercept form for you to read off the slope and $y$-intercept directly.

If an equation is in this form, the slope is the number multiplying $x$.

What is the slope of this equation?

$$y=2^{2}x+3$$

$$y=mx+b$$

In this case, $m=2^2$, which is $4$. The slope is $4$.

You can also use point-slope form to rewrite an equation into slope-intercept form. For example, if the slope is $3$ and the line goes through the point $(2,2)$, start with point-slope form but use $x$ and $y$ for the unknown point:

$$\begin{array}{rl}(y-y_1)& =m\ast (x-x_1)\\ (y-2)& =3\ast (x-2)\end{array}$$

Now rearrange to look like slope-intercept form:

$$\begin{array}{rl}y-2& =3x-6\\ y& =3x-4\end{array}$$

Now you have the equation of the line, and you can see the $y$-intercept is $-4$.

Key points

Linear equation. A linear equation is one in which the highest exponent of a variable is $1$.

Standard form. $Ax+By=C$ where $A$, $B$, and $C$ are integers. Find $x$-intercept by setting $y=0$ and solving for $x$. Find $y$-intercept by setting $x=0$ and solving for $y$. Slope $=-A/B$.

Point-slope form. $(y_2-y_1)=m(x_2-x_1)$ where $m$ is the slope. Find the slope by remembering “rise over run.” Use $(y-y_1)=m(x-x_1)$ if you know the slope and one point to get the equation into slope-intercept form.

Slope-intercept form. $y=mx+b$ where $m=\text{slope}$, $b=y$-intercept.