Standard form of linear equations

A linear equation is an equation that produces a straight line on a graph, as opposed to other equations that produce curves. The three types of linear equations that will be addressed in this chapter are standard form, point-slope form, and slope-intercept form. You can recognize an equation as linear if its highest exponent on a variable is $1$.

Standard form

The standard form of a linear equation is:

$$Ax+By=C$$

where $x$ and $y$ are variables, and $A$, $B$, and $C$ are any integers. This form is useful because we can solve for the $x$-intercept and $y$-intercept easily. To find the $x$-intercept, we set $y=0$ and solve for $x$. If we want to find the $y$-intercept, we set $x=0$ and solve for $y$. One way to remember this is that you should solve for the same variable for which you want the intercept. If you want the $y$-intercept, then you will need to solve for $y$.

We can also use this form to find the slope with the formula $m=-A/B$ where m is the slope of the line. If we do not want to solve the equation algebraically to find the $y$-intercept, we can simply use the formula $C/B$ to quickly find this value.

We will explore a few examples to ensure that we can recognize the key points of a linear equation and find the $x$ and $y$ 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. There is a minus sign in place of the usual addition sign, but it is still linear.

$x$-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 to the next section. You will notice that even if an equation is not in standard form, it can still be linear because we can turn 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 may have a graphed line with two points on it, two coordinate points given in a problem, or even just an equation that you could use to find two points. A linear equation in point-slope form looks like

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

Where $m$ is the slope, the variables with the subscript of $1$ are the coordinates of the first point, and the variables with the subscript of $2$ are the coordinates of the second point. This comes from the formula for slope, which you may be more familiar with,

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

You can more easily remember this formula by the phrase “rise over run.” This means that to find the slope of a line, you count how far the line “rises” in the $y$-direction, and then divide that by how far the line “runs” in the $x$-direction. $(y_2-y_1)$ is the “rise” and $(x_2-x_1)$ is the “run.” 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, $m=\frac{3}{2}$

You may notice that doing the rise over run method actually caused us to perform the full point-slope formula. At the end of the day, either way you remember this equation will work perfectly well.

Find the slope of the following line:

Slope-intercept form

The third form of linear equations we will talk about is slope-intercept form. It is very useful because it helps us determine the slope and the $y$-intercept just by looking at the equation. However, the equation must look a specific way in order for it to work. We call this the slope-intercept form, and it looks like $y=mx+b$, where $y$ and $x$ are variables, $m$ represents the slope, and $b$ represents the $y$-intercept. It is slightly confusing that we choose to use variables $m$ and $b$ instead of $A$ and $B$, but this helps us remember that the equation must be in slope-intercept form in order to determine the slope and the $y$-intercept.

If you see an equation in this form, then you can find the slope to be whatever numbers are attached to $x$. It’s as easy as it sounds. Check out this example.

What is the slope of this equation?

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

$$y=mx+b$$

In this case, $m=2^2$. Or in other words, $4$. The slope is therefore $4$.

We can also use point-slope form as a way to get the equation of a line into slope-intercept form. For example, if we are told that the slope of a line is $3$ and the line goes through the point $(2,2)$, we can use point-slope form to get to slope-intercept form, which will directly tell us the $y$-intercept. To do this, instead of using $y_2$ and $x_2$, we just leave them as $y$ and $x$ in point-slope form.

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

Now, we rearrange the equation to look like slope-intercept form.

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

Now we have the equation of that line, and we can clearly see that 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.