Question

How do you write an equation of a line passing through (-2, -2), perpendicular to `y=x+1`?

Answer 1

See the entire solution process below:

Explanation:

Because the equation given in the problem is in slope-intercept form we can easily determine the slope. The slope-intercept form of a linear equation is: `y = color(red)(m)x + color(blue)(b)`

Where `color(red)(m)` is the slope and `color(blue)(b)` is the y-intercept value.

`y = color(red)(1)x + color(blue)(1)`

Therefore the slope of the line from the problem is: `color(red)(m = 1)`

Let's call the slope of a perpendicular line `m_p`.

The slope of a perpendicular line is: `m_p = -1/m`

Substituting `1` for `m` gives the slope of the perpendicular line in this problem as:

`m_p = -1/1 = -1`

Because we are given a point we can use the point-slope formula to find an equation for the line. The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

Substituting the slope we calculated and the point from the problem gives:

`(y - color(red)(-2)) = color(blue)(-1)(x - color(red)(-2))`

Solution 1) `(y + color(red)(2)) = color(blue)(-1)(x + color(red)(2))`

We can also substitute the values from the problem and the slope we calculated into the slope-intercept formula and solve for `b` to find the equation in slope-intercept form:

`-2 = (color(red)(-1) xx -2) + color(blue)(b)`

`-2 = 2 + color(blue)(b)`

`color(red)(-2) - 2 = color(red)(-2) - 2 + color(blue)(b)`

`-4 = 0 + color(blue)(b)`

`-4 = color(blue)(b)`

`y = color(red)(-1)x - color(blue)(4)`

Solution 2) `y = color(red)(-)x - color(blue)(4)`