Question

What is the equation of the line that is perpendicular to -x+y=7 and passes through (-1,-1)?

a.-x+y=2
b.-x-y=2
c.x-y=2
d.-x-y=0
e. x+y=0

Answer 1

`-x - y = 2`.

Explanation:

If the slope of a line in a plane is
`m` and the
`y`-intercept of that line is
`b`, the slope-intercept equation of that line would be
`y = m\, x + b`.

Rearrange the equation of the given line
`-x + y = 7` in the slope-intercept form to find the slope of this line:

`-x + y = 7`.

`x -x + y = x + 7`.

`y = x + 7`.

Notice that in the slope-intercept equation of this given line, the coefficient of
`x` is
`1`. Thus, the slope of the given line would be
`m_(1) = 1`.

Two lines in a plane are perpendicular to one another if the product of their slopes is
`(-1)`. In other words, if
`m_(1)` and
`m_(2)` are the slopes of two lines perpendicular to each other, then
`m_(1)\, m_(2) = (-1)`.

Since
`m_(1) = 1` for the given line, the slope of the line perpendicular to this given line would be:

`m_(2) = (-1) / m_(1) = (-1) / 1 = -1`.

If the slope of a line in a plane is
`m`, and that line goes through the point
`(x_(0),\, y_(0))`, the equation of that line in point-slope form would be:

`y - y_(0) = m\, (x - x_(0))`.

The slope of the line in question is
`m = (-1)`. It is given that this line goes through the point
`(-1,\, -1)`, where
`x_(0) = (-1)` and
`y_(0) = (-1)`. Thus, the equation of this line in point-slope form would be:

`y - y_(0) = m\, (x - x_(0))`.

`y - (-1) = (-1)\, (x - (-1))`.

`y + 1 = -(x + 1)`.

Rearrange this equation to match the format of the choices:

`y + 1 = -x - 1`.

`x + y = -2`.

`-x - y = 2`.