Question

Find the equation for each line as described. Helpful Hint: A parallel line will have the same slope, a perpendicular line will have a slope that is the opposite reciprocal. After determining slope, use the y-intercept form and the given point to determine the y-intercept, and complete the equation.

1. A line passes through (4, -1) and is perpendicular to y=2x-7
2. A line passes through (2, 4) and is parallel to y = x.
3. A line passes through (2,2) and is perpendicular to y = x
4. A line passes through (-1, 5) and is parallel to y=-x+10

Answer 1

1. A line passes through (4, -1) and is perpendicular to y=2x-7

The slope of the given line is 2. Since the line we are looking for is perpendicular, the slope of the new line will be the opposite reciprocal of 2, which is -1/2.

Now, we'll use the point-slope form to find the equation of the line:

y - y1 = m(x - x1)

y - (-1) = -1/2(x - 4)

y + 1 = -1/2x + 2

y = -1/2x + 1

1. A line passes through (2, 4) and is parallel to y = x.

The slope of the given line is 1. Since the line we are looking for is parallel, the slope of the new line will also be 1.

y - 4 = 1(x - 2)

y - 4 = x - 2

y = x + 2

1. A line passes through (2,2) and is perpendicular to y = x

The slope of the given line is 1. Since the line we are looking for is perpendicular, the slope of the new line will be the opposite reciprocal of 1, which is -1.

y - 2 = -1(x - 2)

y - 2 = -x + 2

y = -x + 4

1. A line passes through (-1, 5) and is parallel to y=-x+10

The slope of the given line is -1. Since the line we are looking for is parallel, the slope of the new line will also be -1.

y - 5 = -1(x - (-1))

y - 5 = -1(x + 1)

y - 5 = -x - 1

y = -x + 4

Explanation:

Answer 2

1. The given line has a slope of 2, so a line perpendicular to it will have a slope of -1/2 (the opposite reciprocal). Using the point-slope form of a line, the equation of the line passing through (4, -1) with a slope of -1/2 is:

y - (-1) = (-1/2)(x - 4)

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

y = (-1/2)x + 1

2. The given line has a slope of 1, so a line parallel to it will also have a slope of 1. Using the point-slope form of a line, the equation of the line passing through (2, 4) with a slope of 1 is:

y - 4 = 1(x - 2)

y - 4 = x - 2

y = x + 2

3. The given line has a slope of 1, so a line perpendicular to it will have a slope of -1 (the opposite reciprocal). Using the point-slope form of a line, the equation of the line passing through (2, 2) with a slope of -1 is:

y - 2 = -1(x - 2)

y - 2 = -x + 2

y = -x + 4

4. The given line has a slope of -1, so a line parallel to it will also have a slope of -1. Using the point-slope form of a line, the equation of the line passing through (-1, 5) with a slope of -1 is:

y - 5 = -1(x - (-1))

y - 5 = -x - 1

y = -x + 4

Hope this helps!