Math Problem Solver

Standard Form:

Perpendicular lines are two lines in the same plane which intersect at a 90 degree angle. Perpendicular lines have slopes that are negative reciprocals of each other, so the slope of the given line can be used to find the slope of a line perpendicular to it.$$\begin{aligned}Ax + By + C &= 0 \\ -5x +2y -6 &= 0 \end{aligned}$$$$\begin{aligned} \text{Slope} &= -\frac{A}{B} \\ \text{Slope} &= \frac{5}{2} \end{aligned}$$The negative reciprocal of $\frac{5}{2}$ is $- \frac{2}{5}$, therefore $A$ and $B$ of a perpendicular line are $2$ and $5$.$$\begin{aligned} 2x +5y + C &= 0 \\ 2(3) +5(-5)+C &= 0 \\ 6 + C &= 25 \\ C &= 19 \end{aligned}$$An equation of the line in standard form is: $2x +5y +19 = 0$.

Slope-Intercept Form:

Perpendicular lines are two lines in the same plane which intersect at a 90 degree angle. Perpendicular lines have slopes that are negative reciprocals of each other, so the slope of the given line can be used to find the slope of a line perpendicular to it.$$\begin{aligned} y = mx+b\end{aligned}$$$$\begin{aligned} y &= \frac{5 x}{2} + 3 \\ \text{Slope} &= m = \frac{5}{2} \end{aligned}$$The negative reciprocal of $\frac{5}{2}$ is $- \frac{2}{5}$, therefore $m$ of a perpendicular line will be $- \frac{2}{5}$.$$\begin{aligned}y &= - \frac{2}{5}x+b \\ -5 &= - \frac{2}{5}(3)+b \\ b &= - \frac{19}{5} \end{aligned}$$The slope-intercept form of the line is: $y = - \frac{2}{5}x - \frac{19}{5}$.