Question

A line is perpendicular to the line `y=2x+3` and has the same `x`-intercept as `x+3y+10=0`. What is the equation of his line?

Answer 1

Step 1: Determine the slope of the second line

Perpendicular means the slope of the line is the negative reciprocal of the other.

Since we're in `y = mx + b`, the slope is at `m`, and `m` is equal to `2`. The negative reciprocal of `2` is `-1/2`. Thus, the line perpendicular to `y = 2x + 3` will have a slope of `-1/2`.

Step 2: Determine the x intercept of `x + 3y + 10 = 0`

The x intercept occurs when `y = 0`. We can therefore state that:

`x + 3(0) + 10 = 0`

`x + 10 = 0`

`x = -10`

Hence, the x intercept is at `(-10, 0)`.

Step 3: Determine the equation of the line using point-slope form

Point slope form is `y - y_1 = m(x - x_1)`, where `(x_1, y_1)` is a point on the linear function and `m` is the slope. We know a point `(-10, 0)` and the slope `-1/2`. We can now compute the equation of the line.

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

`y - 0 = -1/2(x - (-10))`

`y = -1/2x - 5`

Step 4: Declare our answer

The equation of the line perpendicular to `y = 2x + 3` and that has the same x intercept as `x + 3y + 10 = 0` is `y = -1/2x - 5`

Hopefully this helps!