# How do you write the equation of a line in slope intercept, point slope and standard form given (3,-2) and has the slope of 3/4?

Feb 19, 2017

$y = \frac{3}{4} x - \frac{17}{4} , y + 2 = \frac{3}{4} \left(x - 3\right) , 3 x - 4 y = 17$

#### Explanation:

The equation of a line in $\textcolor{b l u e}{\text{point-slope form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y - {y}_{1} = m \left(x - {x}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where m represents the slope and $\left({x}_{1} , {y}_{1}\right) \text{ a point on the line}$

$\text{here "m=3/4" and } \left({x}_{1} , {y}_{1}\right) = \left(3 , - 2\right)$

substituting these values into the equation gives.

$y - \left(- 2\right) = \frac{3}{4} \left(x - 3\right)$

$\Rightarrow y + 2 = \frac{3}{4} \left(x - 3\right) \leftarrow \textcolor{red}{\text{ in point-slope form}}$

The equation in $\textcolor{b l u e}{\text{slope-intercept form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = m x + b} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where m represents the slope and b, the y-intercept.

distribute and simplify the point-slope equation into this form.

$\Rightarrow y + 2 = \frac{3}{4} x - \frac{9}{4}$

$y = \frac{3}{4} x - \frac{9}{4} - 2$

$\Rightarrow y = \frac{3}{4} x - \frac{17}{4} \leftarrow \textcolor{red}{\text{ in slope-intercept form}}$

The equation in $\textcolor{b l u e}{\text{standard form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where A is a positive integer and B ,C are integers.

Rearrange the slope-intercept equation into this form.

$\frac{3}{4} x - y = \frac{17}{4} \leftarrow \text{ multiply through by 4}$

$\Rightarrow 3 x - 4 y = 17 \leftarrow \textcolor{red}{\text{ in standard form}}$