**Introduction**

In this lesson, you will learn how to convert any linear equation to the slope-intercept form of a line.

This video illustrates the lesson material below. Watching the video is optional.

**Convert Linear to Slope-Intercept**

You can use the skills of solving for a variable to change any linear equation into slope-intercept form. All that is required is that you solve for *y* then arrange them so that the term with *x* in it comes first.

**Linear equation**: an equation that is a straight line when graphed.

Here are four different equations:

- \(2x+3y=12\)
- \(4y=16\)
- \(-x=3y-9\)
- \(2=4y-12x\)

Each of these equations represent a straight line; in other words, each of these is a linear equation, but they all look a little bit different. It’s hard to tell what the slope of the line is or what their intercepts are just by looking at them. To easily see the slope and *y*-intercept, convert each one into the slope-intercept form of a line: \begin{align*} y=mx+b\end{align*} where *m* is the slope and *b* is the y-intercept.

Each of the equations listed above can be changed into slope-intercept form by solving for *y*.

**Example 1**

Convert the following linear equation to slope-intercept form: \(2x+3y=12\).

\begin{align*} 2x+3y &=12 &\color{red}\small\text{Solve for y}\\\\ 2x +3y \color{red}\text{ - 2x} &=12 \color{red}\text{ - 2x} &\color{red}\small\text{Subtract 2x from both sides}\\\\ 3y &=12 - 2x &\color{red}\small\text{Left side: 2x's will cancel}\\\\ \frac{3y}{\color{red}\text{3}} &=\frac{12 - 2x} {\color{red}\text{3}} &\color{red}\small\text{Divide both sides by 3}\\\\ y &=\frac{12}{3} \frac{- 2x}{3}&\color{red}\small\text{Left side: the 3's cancel out}\\\\ y &=4 -\frac{2x}{3}&\color{red}\small\text{Right side: simplify the fractions}\\\\ y &=-\frac{2x}{3} + 4&\color{red}\small\text{Rewrite in slope-intercept form} \end{align*}

In this form, you can instantly identify what the slope of the equation is and what the y-intercept is. The slope is \(-\frac{2}{3}\) and the y-intercept is 4.

You want to isolate the *y* all by itself on the left-hand side of the equation. In reality, it doesn’t matter if you isolate it on the left-hand side or the right-hand side as long as everything else is equal to y, but if you isolate it on the left, it will look exactly like the slope-intercept form equation: \(y=mx+b\).

**Example 2**

Convert the following linear equation to slope-intercept form: \(4y=16\).

\begin{align*} 4y &= 16 &\color{red}\small\text{Solve for y}\\\\

\frac{4y}{\color{red}\text{4}} &=\frac{16} {\color{red}\text{4}} &\color{red}\small\text{Divide both sides by 4}\\\\ y &=\frac{16}{4} &\color{red}\small\text{Left side: the 4's cancel out}\\\\ y &=4 &\color{red}\small\text{Slope-intercept form} \end{align*}

The final equation is \(y=4\). This doesn’t look exactly like \(y=mx+b\) but it is. In this case, the *m* is just zero, so you could rewrite this as \(y=0x+4\).

**Example 3**

Convert the following linear equation to slope-intercept form: \(-x=3y-9\).

\begin{align*} -x &=3y-9 &\color{red}\small\text{Solve for y}\\\\ -x \color{red}\text{ +9} &=3y - 9 \color{red}\text{ + 9} &\color{red}\small\text{Add 9 to both sides}\\\\ -x + 9 &=3y &\color{red}\small\text{Right side: 9's will cancel}\\\\ \frac{-x+9}{\color{red}\text{3}} &=\frac{3y} {\color{red}\text{3}} &\color{red}\small\text{Divide both sides by 3}\\\\ \frac{-x}{3} +\frac{9}{3} &=y &\color{red}\small\text{Left side: the 3's cancel out}\\\\ -\frac{1}{3}x + 3& = y&\color{red}\small\text{Left side: simplify the fractions}\\\\ y&=-\frac{1}{3}x + 3 &\color{red}\small\text{Rewrite in slope-intercept form} \end{align*}

\(-\frac{1}{3}x+3=y\) is the same as saying \(y=-\frac{1}{3}x+3\), which is in the \(y=mx+b\) form. The slope is \(-\frac{1}{3}\), and the y-intercept is 3.

**Example 4**

Convert the following linear equation to slope-intercept form: \(2=4y-12x\). This is the same as \(4y-12x=2\).

\begin{align*} 4y -12x &=2 &\color{red}\small\text{Solve for y}\\\\ 4y -12x \color{red}\text{ + 12x} &=2 \color{red}\text{ + 12x} &\color{red}\small\text{Add 12x to both sides}\\\\ 4y &=12x + 2 &\color{red}\small\text{Left side: 12x's will cancel}\\\\ \frac{4y}{\color{red}\text{4}} &=\frac{12x + 2} {\color{red}\text{4}} &\color{red}\small\text{Divide both sides by 4}\\\\ y &=\frac{12}{4}x + \frac{2}{4}&\color{red}\small\text{Left side: the 4's cancel out}\\\\ y &= 3x + \frac{1}{2}&\color{red}\small\text{Right side: simplify the fractions} \end{align*}

\(y=3x+\frac{1}{2}\) is already in slope-intercept form. From this form, you see that the slope is 3 and the equation crosses the y-axis at \(\frac{1}{2}\).

**Things to Remember**

- Just because an equation isn’t in slope-intercept form doesn’t mean that it can’t be solved for
*m*and*b*.- Rearrange the equation to isolate or solve for
*y*.

- Rearrange the equation to isolate or solve for
- Remember to multiply by the multiplicative inverse to undo the fractions by the variable.
- Check your work after each step!

### Practice Problems

**Change the following equations into the slope-intercept form of a line:**

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