Question

hi ,how do i find an equation of the line that passes through the point (2, −5) and is perpendicular to the line passing through the points (−3, −2) and (5, 4). (Let x be the independent variable and y be the dependent variable.)

Answer 1

EXPLANATION

Since we need a line that passes through the points (2,-5) and perpendicular to the line that passes through the points (x_1,y_1) = (−3, −2) and (x_2,y_2) = (5, 4).

We first need to compute the slope of this line by applying the slope formula:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Substituting terms:

$\text{Slope}=\frac{4-(-2)}{5-(-3)}$

Subtracting numbers:

$\text{Slope}=\frac{6}{8}=\frac{3}{4}$

As the first line is perpendicular to the second, the slope should be negative and reciprocal.

Thus, the slope of the first line is slope= -4/3

Now, we already know that the equation of a line is as follows:

$y=mx\text{ + b}$

Where x=2 and y=-5 (the given point) and m=slope= -4/3

Plugging in the values into the equation:

$-5=-\frac{4}{3}\cdot2+b$

Multiplying numbers:

$-5\text{ = -}\frac{8}{3}+b$

Adding +8/3 to both sides:

$-\frac{7}{3}=b$

Finally, the equation of the line is the following:

$y=-\frac{4}{3}x-\frac{7}{3}$