Question

write the slope - intercept form of the equation of the line described
1 through (-2,2)
perpendicular to y = \frac{2}{3}x - 2
2 through (2,-1)
perpendicular to y=-5x + 1

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the line $y = \frac{2}{3}x - 2$ is $m_1=\frac{2}{3}$. If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the required line be $m$. Then $m\times\frac{2}{3}=-1$, so $m =-\frac{3}{2}$.

Step2: Use the point - slope form to find the equation

The point - slope form of a line is $y - y_1=m(x - x_1)$. The line passes through the point $(-2,2)$ and $m =-\frac{3}{2}$. Substitute $x_1=-2,y_1 = 2,m=-\frac{3}{2}$ into the point - slope form: $y - 2=-\frac{3}{2}(x + 2)$.

Step3: Convert to slope - intercept form

Expand the right - hand side: $y-2=-\frac{3}{2}x-3$. Add 2 to both sides of the equation: $y=-\frac{3}{2}x-1$.

For the second part:

Step1: Find the slope of the perpendicular line

The slope of the line $y=-5x + 1$ is $m_1=-5$. If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the required line be $m$. Then $m\times(-5)=-1$, so $m=\frac{1}{5}$.

Step2: Use the point - slope form to find the equation

The line passes through the point $(2,-1)$ and $m=\frac{1}{5}$. Substitute $x_1 = 2,y_1=-1,m=\frac{1}{5}$ into the point - slope form $y - y_1=m(x - x_1)$: $y+1=\frac{1}{5}(x - 2)$.

Step3: Convert to slope - intercept form

Expand the right - hand side: $y + 1=\frac{1}{5}x-\frac{2}{5}$. Subtract 1 from both sides: $y=\frac{1}{5}x-\frac{2}{5}-1=\frac{1}{5}x-\frac{7}{5}$.

Answer:

1. $y =-\frac{3}{2}x-1$
2. $y=\frac{1}{5}x-\frac{7}{5}$