Question

Select the equation that is perpendicular to y = x+4 and goes through (2,2)

a
y=x+4

b
y=-x-2

c
y=-x+4

d
y=x+2

Answer 1

To find the equation that is perpendicular to \( y = x + 4 \) and passes through the point \( (2, 2) \), we first need to determine the slope of the given line.

The equation \( y = x + 4 \) can be rewritten in slope-intercept form \( y = mx + b \), where \( m \) is the slope. From this, we see that the slope \( m \) of the line is \( 1 \).

For a line to be perpendicular to another, the slopes must multiply to \(-1\). Therefore, the slope \( m_{\text{perpendicular}} \) of the line we are looking for will be:

\[
m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{1} = -1
\]

Now we know the slope of the line we are looking for is \(-1\). We can use the point-slope form of the equation of a line, which is given by:

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

Where \( (x_1, y_1) = (2, 2) \) and \( m = -1 \). Substituting these values in:

\[
y - 2 = -1(x - 2)
\]

Simplifying this gives:

\[
y - 2 = -x + 2
\]

\[
y = -x + 4
\]

Now we can compare this with the options provided:

a) \( y = x + 4 \)

b) \( y = -x - 2 \)

c) \( y = -x + 4 \)

d) \( y = x + 2 \)

The correct equation that is perpendicular to \( y = x + 4 \) and passes through the point \( (2, 2) \) is:

**c) \( y = -x + 4 \)**.