Question

write the equation of the line through the given point. use slope - intercept form. (-3,1); perpendicular to y = -\frac{4}{3}x + 3 write an equation for the line in slope - intercept form. (simplify your answer. use integers or fractions for any numbers in the equation.)

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line is $-\frac{4}{3}$. For two perpendicular lines with slopes $m_1$ and $m_2$, $m_1\times m_2=- 1$. Let the slope of the required line be $m$. Then $-\frac{4}{3}\times m=-1$, so $m = \frac{3}{4}$.

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

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(-3,1)$ and $m=\frac{3}{4}$. Substitute these values: $y - 1=\frac{3}{4}(x+3)$.

Step3: Convert to slope - intercept form

Expand the right - hand side: $y - 1=\frac{3}{4}x+\frac{9}{4}$. Add 1 to both sides: $y=\frac{3}{4}x+\frac{9}{4}+1=\frac{3}{4}x+\frac{9 + 4}{4}=\frac{3}{4}x+\frac{13}{4}$.

Answer:

$y=\frac{3}{4}x+\frac{13}{4}$