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Convert Any Linear Equation to Slope-Intercept Form of a Line
> ... Math > Slopes, Intercepts, Equation of a Straight Line > Convert Any Linear Equation to Slope-Intercept Form of a Line

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.
  • 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:

1. \({\text{y}}+14=-4{\text{x}}\) (
Solution
x
Solution: \({\text{y}}=-4{\text{x}}-14\)
)2. \({\text{y}}{-}7 = \dfrac{1}{3}{\text{x}}\) (
Solution
x
Solution: \({\text{y}}=\dfrac{1}{3}{\text{x}}+7\)
)3. \({\text{y}} + \dfrac{3}{8} = \dfrac{1}{8}{\text{x}}\) (
Solution
x
Solution: \({\text{y}}=\dfrac{1}{8}{\text{x}}-\dfrac{3}{8}\)
)4. \({\text{x}}=\dfrac{{\text{y}}+36}{9}\) (
Solution
x
Solution: \({\text{y}}=9{\text{x}}-36\)

Details:
To change the equation into slope-intercept form, write it in the form \(y=mx+b\).

Start with the original equation:

\({\text{x}}=\dfrac{{\text{y}}+36}{9}\)

You want to isolate the y, so the first step is to multiply both sides by 9.

\(9{\text{x}}=\left(\dfrac{{\text{y}}+36}{9}\right)9\)

Then cancel out the 9’s on the right side.

\(9{\text{x}}=\left(\dfrac{{\text{y}}+36}{\:\cancel{9}}\right)\:\cancel{9}\)

Which gives you:

\({\text{9x=y+36}}\)

Then subtract 36 from both sides.

\(9{\text{x}} {\color{Red}{-}36} = {\text{y}} + 36 {\color{Red}{-}36}\)

Which simplifies to:

\(9{\text{x}}{-}36={\text{y}}\)

Which is the same equation as:

\({\text{y}}=9{\text{x}}-36\)
)5. \(-6{\text{x}}{-}2{\text{y}}=-7\) (
Solution
x
Solution: \({\text{y}}=-3{\text{x}}+\dfrac{7}{2}\)

Details:
To change the equation into slope-intercept form, write it in the form \(y=mx+b\).

Start with the original equation:

\(-6{\text{x}}{-}{2}\text{y}=-7\)

You want to isolate the term with \(y\), so you add \(6x\) to both sides:

\(-6\text{x}+6{\text{x}}{-}{2}\text{y}=-7+6\text{x}\)

Then simplify to:

\(-2\text{y}=-7+6\text{x}\)

Now you can multiply both sides by \({\color{Red} -\dfrac{1}{2}}\) (or divide by \(−2\), which is an equivalent operation).

\({\color{Red} -\dfrac{1}{2}}\left ( -2\text{y} \right )= {\color{Red} -\dfrac{1}{2}}\left ( -7+6\text{x} \right )\)

Multiply on the left and distribute on the right to get:

\({\color{Red}1}\text{y}= {\color{Red} -\dfrac{1}{2}}\left (-7 \right )+{\color{Red} -\dfrac{1}{2}}\left (6\text{x} \right )\)

Which gives you:

\({\text{y}}=\dfrac{7}{2}-\dfrac{6}{2}{\text{x}}\)

Which simplifies to:

\({\text{y}}=\dfrac{7}{2}-{\color{Red} 3}{\text{x}}\)

Then you can rewrite the equation so that it is in slope-intercept form:

\({\text{y}}=-3{\text{x}}+\dfrac{7}{2}\)
)6. \(3{\text{x}}{-}2{\text{y}}=-1\) (
Solution
x
Solution: \({\text{y}}=\dfrac{3}{2}{\text{x}}+\dfrac{1}{2}\)
)

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