# Find the equation of a line that goes through point (100,-25) and is perpendicular to y=-25?

Jul 23, 2018

$x = 100$

#### Explanation:

$y = - 25$

$\text{is a horizontal line parallel to the x-axis and passing}$
$\text{through all points in the plane with a y-coordinate } - 25$

$\text{a line perpendicular to it must be a vertical line}$

$\text{the equation of such a line is } x = c$

$\text{where c is the value of the x-coordinate the line}$
$\text{passes through}$

$\text{here the point is } \left(\textcolor{red}{100} , - 25\right)$

$\text{the equation of the perpendicular line is } x = 100$

Jul 24, 2018

$y = \frac{1}{25} x - 725$

#### Explanation:

Recall the standard equation of a line is;

$y = m x + c$

Also recall;

When its perpendicular;

${m}_{1} \cdot {m}_{2} = - 1$

For the first equation;

Coordinate, $\left(100 , - 25\right)$

${x}_{1} = 100$

${y}_{1} = - 25$

${m}_{1} = - 25$

${m}_{1} \cdot {m}_{2} = - 1$

${m}_{2} = - \frac{1}{m} _ 1$

Plugging in the value;

${m}_{2} = - \frac{1}{- 25}$

${m}_{2} = \frac{1}{25}$

Now the new equation of a line is;

$\frac{y - {y}_{1}}{x - {x}_{1}} = {m}_{2}$

Plugging in the values;

$\frac{y - \left(- 25\right)}{x - 100} = \frac{1}{25}$

$\frac{y + 25}{x - 100} = \frac{1}{25}$

Cross multiplying;

$\left(y + 25\right) 25 = 1 \left(x - 100\right)$

$25 y + 625 = x - 100$

Collecting like terms;

$25 y = x - 100 - 625$

$25 y = x - 725$

$y = \frac{x}{25} - 725 \to \text{Equation}$

$y = \frac{1}{25} x - 725$