Question

Find the line that is perpendicular to y=x+2 and passes through the point (3,0) .(1 point) Responses y=x−3 y is equal to x minus 3 y=12x−32 y is equal to 1 half x minus 3 halves y=−x+3 y is equal to negative x plus 3 y=−2x+6 y is equal to negative 2 x plus 6

Answer 1

To determine the line that is perpendicular to \(y = x + 2\) and passes through the point \((3, 0)\), we first need to find the slope of the given line.

The equation \(y = x + 2\) can be expressed in slope-intercept form \(y = mx + b\) where \(m\) is the slope. Here, the slope \(m\) is \(1\) (since the coefficient of \(x\) is \(1\)).

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope of the line we are looking for is:

\[
m = -\frac{1}{1} = -1
\]

Now, we use the point-slope form of the equation of a line, which is given by:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. Substituting the point \((3, 0)\) and the slope \(-1\):

\[
y - 0 = -1(x - 3)
\]

This simplifies to:

\[
y = -x + 3
\]

Thus, the line that is perpendicular to \(y = x + 2\) and passes through the point \((3, 0)\) is:

\[
y = -x + 3
\]

So, the correct response is:

**y = −x + 3**.