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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 $1$.

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.

$y+3=3x$

(spoiler)

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

This is a linear equation. The form looks very different, but we can rearrange it using algebra so that it appears $3x−y=3$.

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

$0+3=3x$

Solve for $x$:

$x=3/3$

The $x$-intercept is $1$.

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

$y+3=3(0)$

Solve for $y$:

$y=−3$

The $y$-intercept is $−3$.

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

$4−x_{2}=y$

(spoiler)

Is the highest exponent on the variables $1$? No, the highest power is $x_{2}$.

This is **not** a linear equation. We will not solve for the intercepts.

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,

$slope=(x_{2}−x_{1})(y_{2}−y_{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 $(3,5)$?

How far does the line “rise” from the first point to the second point? $5−2=3=rise$.

How far does the line “run” from the first point to the second point? $3−1=2=run$.

Slope is rise over run, $m=23 $

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:

(spoiler)

Rise over run

$rise=(1−(−3))=4$

$run=(3−(−1))=4$

$runrise =44 =1$

The slope of this line is $1$.

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.

$(y−y_{1})(y−2) =m∗(x−x_{1})=3∗(x−2) $

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

$y−2y =3x−6=3x−4 $

Now we have the equation of that line, and we can clearly see that the $y$-intercept is $−4$.

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