Question

(a) Write an equation of the line that passes through the point and is parallel to

the given line.
(b) Write an equation of the line that passes through the point and is perpendicular to the given line.

Answer 1

To find equations of lines parallel or perpendicular to a given line, use the same slope for a parallel line and the negative reciprocal of the slope for a perpendicular line. Apply the point that the line must pass through to determine the specific y-intercept for each case.

Step-by-step explanation:

To solve this problem, one must understand the concept of the slope and y-intercept of a linear equation. Given the equation of a line in slope-intercept form, y = a + bx, we know that 'b' represents the slope, and 'a' represents the y-intercept. When finding a line parallel to a given line, the slope must be the same, while the y-intercept may differ. To find a perpendicular line, one must use the negative reciprocal of the given line's slope.

For example, if we are given a line with the equation y = 2 + 3x and we want a line that is parallel through point (1, 4), the new line will have the same slope, 3, so its equation will start with y = 3x + c. Substituting the point (1, 4) into this equation gives us 4 = 3(1) + c, which after solving gives c = 1. So the equation of the parallel line is y = 3x + 1.

To find a perpendicular line through the same point (1, 4), we use the negative reciprocal of 3, which is -1/3. The equation starts as y = -1/3x + d. Substituting the point (1, 4) into this equation gives us 4 = -1/3(1) + d, which after solving gives d = 13/3. So, the perpendicular line's equation is y = -1/3x + 13/3.

Answer 2

For the equation of a parallel line:

Parallel lines will always share the same slope of the original equation, although they may pass through different points.

You can use the point-slope formula to write this new equation:

y - y₁ = m (x - x₁), where

x₁ and y₁ are the points the line must pass and m is the slope

First, find the slope of the original line:

Pick 2 points from the original line and plug them into the slope equation:

(-3,0) (-1,2)

y₂ - y₁/ x₂ - x₁ :
`(2- 0)/(-1- (-3))` =
`(2)/(2)` = 1

m, or the slope,= 1

(x₁, y₁) = (1, -2) or the points the line must pass

Now, plug in your information into the equation to find the parallel line:

y - y₁ = m (x - x₁)

y- (-2) = 1 (x - 1)

y + 2 = x-1

y = x - 3

This is the equation for the parallel line.

For the perpendicular line, the process is the same. However, the slope for a perpendicular line will always be the negative reciprocal of the original slope.

The original slope is 1. The negative reciprocal of 1 is -1

The equation for the perpendicular line is y = -x - 3