Question

What passes through (-8,6) and is parallel to y=2x+6

What passes through (4,-1) and is perpendicular to y=3x-1

What passes through (9,-5) and is vertical

What passes through (-2,7) and is horizontal

Answer 1

1. What passes through (-8,6) and is parallel to
`\displaystyle\sf y=2x+6`

To find a line parallel to
`\displaystyle\sf y=2x+6` passing through
`\displaystyle\sf (-8,6)`, we can use the slope-intercept form of a line
`\displaystyle\sf y=mx+b`, where
`\displaystyle\sf m` is the slope and
`\displaystyle\sf b` is the y-intercept.

Since the given line
`\displaystyle\sf y=2x+6` has a slope of
`\displaystyle\sf 2`, any line parallel to it will have the same slope. Therefore, the line passing through
`\displaystyle\sf (-8,6)` and parallel to
`\displaystyle\sf y=2x+6` can be represented as
`\displaystyle\sf y=2x+b`, where
`\displaystyle\sf b` is the y-intercept we need to determine.

To find
`\displaystyle\sf b`, substitute the coordinates of the point
`\displaystyle\sf (-8,6)` into the equation:

`\displaystyle\sf 6=2(-8)+b`

`\displaystyle\sf 6=-16+b`

`\displaystyle\sf b=6+16`

`\displaystyle\sf b=22`

Therefore, the equation of the line passing through
`\displaystyle\sf (-8,6)` and parallel to
`\displaystyle\sf y=2x+6` is
`\displaystyle\sf y=2x+22`.

2. What passes through (4,-1) and is perpendicular to
`\displaystyle\sf y=3x-1`

To find a line perpendicular to
`\displaystyle\sf y=3x-1` passing through
`\displaystyle\sf (4,-1)`, we know that the slopes of perpendicular lines are negative reciprocals of each other.

The given line
`\displaystyle\sf y=3x-1` has a slope of
`\displaystyle\sf 3`. The negative reciprocal of
`\displaystyle\sf 3` is
`\displaystyle\sf -(1)/(3)`.

Using the point-slope form of a line
`\displaystyle\sf y-y_(1)=m(x-x_(1))`, where
`\displaystyle\sf m` is the slope and
`\displaystyle\sf (x_(1),y_(1))` are the coordinates of a point on the line, we can substitute
`\displaystyle\sf m=-(1)/(3)` and
`\displaystyle\sf (x_(1),y_(1))=(4,-1)` into the equation.

`\displaystyle\sf y-(-1)=-(1)/(3)(x-4)`

`\displaystyle\sf y+1=-(1)/(3)x+(4)/(3)`

`\displaystyle\sf y=-(1)/(3)x+(4)/(3)-1`

`\displaystyle\sf y=-(1)/(3)x+(1)/(3)`

Therefore, the equation of the line passing through
`\displaystyle\sf (4,-1)` and perpendicular to
`\displaystyle\sf y=3x-1` is
`\displaystyle\sf y=-(1)/(3)x+(1)/(3)`.

3. What passes through (9,-5) and is vertical

A vertical line has an undefined slope and is of the form
`\displaystyle\sf x=a`, where
`\displaystyle\sf a` is a constant.

Since the line passes through
`\displaystyle\sf (9,-5)`, the equation of the vertical line can be written as
`\displaystyle\sf x=9`.

Therefore, the equation of the line passing through
`\displaystyle\sf (9,-5)` and is vertical is
`\displaystyle\sf x=9`.

4. What passes through (-2,7) and is horizontal

A horizontal line has a slope of 0 and is of the form
`\displaystyle\sf y=b`, where
`\displaystyle\sf b` is a constant.

Since the line passes through
`\displaystyle\sf (-2,7)`, the equation of the horizontal line can be written as
`\displaystyle\sf y=7`.

Therefore, the equation of the line passing through
`\displaystyle\sf (-2,7)` and is horizontal is
`\displaystyle\sf y=7`.

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Answer 2

To find the equations of lines that pass through given points and have specific slopes or orientations, we can use the point-slope form of a linear equation, which is given by:

y - y₁ = m(x - x₁)

where (x₁, y₁) are the coordinates of the given point, and m is the slope of the line.

Parallel to y = 2x + 6 and passing through (-8, 6):

Since the line we are looking for is parallel to y = 2x + 6, it will have the same slope of 2. Using the point-slope form, we have:

y - 6 = 2(x - (-8))

y - 6 = 2(x + 8)

y - 6 = 2x + 16

y = 2x + 22

Perpendicular to y = 3x - 1 and passing through (4, -1):

The slope of the line perpendicular to y = 3x - 1 will be the negative reciprocal of the slope of y = 3x - 1. The slope of y = 3x - 1 is 3, so the perpendicular line will have a slope of -1/3. Using the point-slope form:

y - (-1) = (-1/3)(x - 4)

y + 1 = (-1/3)(x - 4)

y + 1 = (-1/3)x + 4/3

y = (-1/3)x + 4/3 - 1

y = (-1/3)x + 1/3

Vertical line passing through (9, -5):

A vertical line has an undefined slope since its x-coordinate does not change. Therefore, the equation of the line passing through (9, -5) will be x = 9. This line is parallel to the y-axis.

Horizontal line passing through (-2, 7):

A horizontal line has a slope of 0 since its y-coordinate does not change. Therefore, the equation of the line passing through (-2, 7) will be y = 7. This line is parallel to the x-axis.