Question

Line q has an equation of y+7=9(x+4). Line r is perpendicular to line q and passes through (9, – 10). What is the equation of line r

Answer 1

To find the equation of line r, which is perpendicular to line q, we need to determine the slope of line q and then use it to find the slope of line r.

The equation of line q is y + 7 = 9(x + 4). By rearranging the equation, we can see that the slope of line q is 9.

Since line r is perpendicular to line q, the slope of line r will be the negative reciprocal of the slope of line q. The negative reciprocal of 9 is -1/9.

Now that we have the slope of line r, we can use the point (9, -10) that line r passes through to find the equation of line r using the point-slope form.

Using the point-slope form, the equation of line r is y - (-10) = -1/9(x - 9). Simplifying this equation gives us y + 10 = -1/9(x - 9).

Therefore, the equation of line r is y + 10 = -1/9(x - 9).

Answer 2

`\sf x + 9y = -81`

Explanation:

To find the equation of the line [tex]\sf r [/tex] that is perpendicular to line
`\sf q` and passes through the point (9, -10), we need to determine the slope of line
`\sf q` and then use the negative reciprocal of that slope for line
`\sf r`.

The equation of line
`\sf q` is given as
`\sf y + 7 = 9(x + 4)`.

First, let's rewrite the equation in the slope-intercept form (
`\sf y = mx + b`) where
`\sf m` is the slope and
`\sf b` is the y-intercept.

`\sf y + 7 = 9(x + 4)`

`\sf y + 7 = 9x + 36`

`\sf y = 9x + 29`

Now we can see that the slope of line
`\sf q` is 9.

Since line
`\sf r` is perpendicular to line
`\sf q`, the slope of
`\sf r` will be the negative reciprocal of the slope of
`\sf q`.

`\sf \text{Slope of } r = -(1)/(9)`

Now, we can use the point-slope form of the equation of a line:

`\sf y - y_1 = m(x - x_1)`

where
`\sf (x_1, y_1)` is the given point (9, -10) and
`\sf m` is the slope.

Substitute known value:

`\sf y - (-10) = -(1)/(9)(x - 9)`

`\sf y + 10 = -(1)/(9)x + 1`

Now, we can rearrange it to get the equation in standard form:

`\sf (1)/(9)x + y = -9`

Multiply through by 9 to get rid of the fraction:

`\sf x + 9y = -81`

So, the equation of line
`\sf r` is
`\sf x + 9y = -81`.