Question

What is the equation of the line that goes through `(4,-2)` and is perpendicular to `y=x`?

Answer 1

First, we will find the slope of the said perpendicular line. This is done by taking the slope of the given equation, and finding the opposite reciprocal of it. In this case, the equation `y=x` is the same as `y=1x`, so the given slope would be 1.

Now, we find the opposite reciprocal by putting the given slope over one, as such:

`1/1`

Then, we change the sign, whether from positive to negative, or vice versa. In this case, the given slope is positive,m so we would make it negative, as such:

`(1/1)*-1 = -1/1`

After finding the opposite of the slope, we must find the reciprocal; this is made by swapping the numerator and denominator (having them trade places). Because the given slope is already 1, there will not be a drastic change, as shown below:

`-1/1 = -1/1`

So, the new slope of the perpendicular line is -1

Now that we have the slope, we can use the point-slope equation to find the equation of the new line. The formula is such:

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

where `y_1` and `x_1` are the given coordinates, and `m` is the slope. Now, plugging in the information given, we should be able to solve the problem:

`y-y_1 = m(x-x_1)`
=> `y-(-2) = -1(x-(4))`
=> `y+2 = -1(x-4)`
=> `y+2 = -1x+4`
=> `y = -1x+2`

Final answer: => `y = -1x+2`