Question

I need it step by step please​

Answer 1

Answer:

The equation of the line passes through (3, -2) and parallel to the given line:

• $y = 3x-11$

The equation of the line passes through (3, -2) and perpendicular to the given line:

• $y=-\frac{1}{3}x-1$

Step-by-step explanation:

Given the points on the graph line

(2, -1)

(1, -4)

Finding the slope between (2, -1) and (1, -4)

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

$\left(x_1,\:y_1\right)=\left(2,\:-1\right),\:\left(x_2,\:y_2\right)=\left(1,\:-4\right)$

$m=\frac{-4-\left(-1\right)}{1-2}$

$m=3$

Equation of the line passes through (3, -2) and parallel to the given line.

We know that parallel lines have the same slope.

so the equation of the line parallel to the given line = 3

Thus, using the point-slope form of the line equation

$y-y_1=m\left(x-x_1\right)$

where m is the slope of the line and (x₁, y₁) is the point

substituting the values m = 3 and the point (3, -2)

$y - (-2) = 3(x-3)$

$y+2 = 3x-9$

$y = 3x-9-2$

$y = 3x-11$

Therefore, the equation of the line passes through (3, -2) and parallel to the given line:

$y = 3x-11$

The equation of the line passes through (3, -2) and perpendicular to the given line

We know that a line perpendicular to another line contains a slope that is the negative reciprocal of the slope of the other line, such as:

slope = m = 3

perpendicular slope = – 1/m = -1/3 = -1/3

Therefore, substituting the values of perpendicular slope = -1/3 and the point (3, -2) in the point-slope form of the line equation

$y-\left(-2\right)=-\frac{1}{3}\left(x-3\right)$

$y+2=-\frac{1}{3}\left(x-3\right)$

subtract 2 from both sides

$y+2-2=-\frac{1}{3}\left(x-3\right)-2$

$y=-\frac{1}{3}x-1$

Therefore, the equation of the line passes through (3, -2) and perpendicular to the given line:

$y=-\frac{1}{3}x-1$

Conclusion:

The equation of the line passes through (3, -2) and parallel to the given line:

• $y = 3x-11$

The equation of the line passes through (3, -2) and perpendicular to the given line:

• $y=-\frac{1}{3}x-1$