Question

Find the equation of the line parallel to.
y, equals, minus, one half, x, plus, 2, .y=−
2
1
​
x+2
and going through the point. , left parenthesis, 2, comma, minus, 9, right parenthesis, .
(2,−9)

your answer should be in slope intercept form. , y, equals, m, x, plus, b
y=mx+b

Answer 1

To find the equation of a line that is parallel to the given line, we first need to identify the slope of the given line. The equation of the given line is:

\[
y = -\frac{1}{2}x + 2
\]

From this equation, we can see that the slope \( m \) is \( -\frac{1}{2} \).

Since parallel lines have the same slope, the slope of the line we are looking for will also be \( -\frac{1}{2} \).

Next, we need to find the \( y \)-intercept \( b \) of the new line that goes through the point \( (2, -9) \). We can use the point-slope form of the equation of a line, which can be rearranged into slope-intercept form.

Using the slope \( m = -\frac{1}{2} \) and the point \( (2, -9) \):

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

Substituting \( (x_1, y_1) = (2, -9) \) and \( m = -\frac{1}{2} \):

\[
y - (-9) = -\frac{1}{2}(x - 2)
\]

This simplifies to:

\[
y + 9 = -\frac{1}{2}(x - 2)
\]

Distributing \( -\frac{1}{2} \):

\[
y + 9 = -\frac{1}{2}x + 1
\]

Now, to isolate \( y \), we subtract 9 from both sides:

\[
y = -\frac{1}{2}x + 1 - 9
\]

\[
y = -\frac{1}{2}x - 8
\]

Thus, the equation of the line that is parallel to \( y = -\frac{1}{2}x + 2 \) and goes through the point \( (2, -9) \) in slope-intercept form is:

\[
\boxed{y = -\frac{1}{2}x - 8}
\]