# How do you find a standard form equation for the line with (7,-4) and perpendicular to the line whose equation is x-7y-4=0?

Oct 5, 2017

$7 x + y = 45$

#### Explanation:

$\text{the equation of a line in "color(blue)"standard form}$ 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.

• " given a line with slope m then the slope of a line"
$\text{perpendicular to it is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

$\text{rearrange "x-7y-4=0" into "color(blue)"slope-intercept form}$

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\Rightarrow y = \frac{1}{7} x - \frac{4}{7} \Rightarrow m = \frac{1}{7}$

$\Rightarrow {m}_{\textcolor{red}{\text{perpendicular}}} = - \frac{1}{\frac{1}{7}} = - 7$

$\Rightarrow y = - 7 x + b \leftarrow \text{ partial equation}$

$\text{to find b substitute "(7,-4)" into the partial equation}$

$- 4 = - 49 + b \Rightarrow b = 45$

$\Rightarrow y = - 7 x + 45 \leftarrow \textcolor{red}{\text{ in slope-intercept form}}$

$\Rightarrow 7 x + y = 45 \leftarrow \textcolor{red}{\text{ in standard form}}$

Oct 5, 2017

$7 x + y = 45$

#### Explanation:

The given equation is in standard form which is $a x + b y = c$

but we need to know its slope, so change it into the form $y = m x + c$

$x - 7 y - 4 = 0 \text{ "rarr x-4 =7y" } \rightarrow 7 y = x - 4$

$y = \textcolor{b l u e}{\frac{1}{7}} x - \frac{4}{7}$

$\rightarrow m = \frac{1}{7}$

If lines are perpendicular then ${m}_{1} \times {m}_{2} = - 1$

(One slope is the negative reciprocal of the other - flip and change the sign.)

The slope perpendicular to $\frac{1}{7}$ is $- 7$

Now we have a point $\left(7 , - 4\right)$ and $m = - 7$ so substitute into the point-slope formula for a straight line.

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$y - \left(- 4\right) = - 7 \left(x - 7\right)$

$y + 4 = - 7 x + 49 \text{ } \leftarrow$ change to standard form:

$7 x + y = 49 - 4$

$7 x + y = 45$