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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

We can use the skills of solving for a variable to change any linear equation into slope-intercept form. All that is required is that we 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
Convert Any Linear Equation to Slope-Intercept Form of a Line

Remember, the key to converting a linear equation to slope-intercept form is to solve for $$y$$ using the tools we learned in PC 101 Weeks 11 and 12. Solving for a variable is used when analyzing data.

### Practice Problems

Change the following equations into the slope-intercept form of a line:
1. $${\text{y}}+14=-4{\text{x}}$$ (
Solution
Solution:
$${\text{y}}=-4{\text{x}}-14$$
)
2. $${\text{y}}{-}7 = \dfrac{1}{3}{\text{x}}$$ (
Solution
Solution:
$${\text{y}}=\dfrac{1}{3}{\text{x}}+7$$
)
3. $${\text{y}} + \dfrac{3}{8} = \dfrac{1}{8}{\text{x}}$$ (
Solution
Solution:
$${\text{y}}=\dfrac{1}{8}{\text{x}}-\dfrac{3}{8}$$
)
4. $${\text{x}}=\dfrac{{\text{y}}+36}{9}$$ (
Solution
Solution:
$${\text{y}}=9{\text{x}}-36$$
Details:
To change the equation into slope-intercept form, we write it in the form $$y=mx+b$$.

$${\text{x}}=\dfrac{{\text{y}}+36}{9}$$

We want to isolate the y, so our 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 us:

$${\text{9x=y+36}}$$

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
Solution:
$${\text{y}}=-3{\text{x}}+\dfrac{7}{2}$$
Details:
To change the equation into slope-intercept form, we need to write it in the form

$$y=mx+b$$

$$-6{\text{x}}{-}{2}\text{y}=-7$$

We want to isolate the term with $$y$$, so we 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 we 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 us:

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

Which simplifies to:

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

Then we 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
Solution:
$${\text{y}}=\dfrac{3}{2}{\text{x}}+\dfrac{1}{2}$$
)

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