Question

Part A. Write an equation of a line that is perpendicular to the line y = 2/3x + 5 and passes through the point (4, 0).

Answer 1

Answer:

$y = -\frac{3}{2}x+6$

Step-by-step explanation:

Given:

The given equation of the line $y = \frac{2}{3} x + 5$ that passes through the point (4, 0).

Part A.

The equation of the line.

$y=mx+c$-----------(1)

Where:

m = Slope of the line

c = y-intercept

The given equation of the line.

$y = \frac{2}{3} x + 5$

Comparing the given equation with equation 1.

The slope of the line is $m=\frac{2}{3}$ and y-intercept $c = 5$

We know that the slope of the perpendicular line is $(-\frac{1}{m})$.

So the slope of the perpendicular line is $(m=-\frac{1}{\frac{2}{3}}=-\frac{3}{2})$.

Using point slope formula we write the equation of the perpendicular line that passes through the point (4, 0).

$y-y_{1} = m(x-x_{1}$

Now we substitute the slope of the perpendicular line $m=-\frac{3}{2}$ and $y_{1} = 0, x_{1}=4$ from point (4, 0) in above equation.  

$y-0 = -\frac{3}{2}(x-4)$

$y = -\frac{3}{2}x-(-\frac{3}{2}\times 4)$

$y = -\frac{3}{2}x-(-3\times 2)$

$y = -\frac{3}{2}x-(-6)$

$y = -\frac{3}{2}x+6$

Therefore the perpendicular line equation is $y = -\frac{3}{2}x+6$.

Part B.

1. The slope of the perpendicular line is $m=-\frac{3}{2}$.

2. The y-intercept of the perpendicular line is 6.

3. The equation of the line $y = -\frac{3}{2}x+6$ is perpendicular to the equation of line $y = \frac{2}{3} x + 5$.