To find the equation of a line that is parallel to a given line and has a specific x-intercept, you need to follow these steps:
1. Identify the slope of the given line:
A line parallel to another will have the same slope. The equation given is [tex]\( y = \frac{2}{3} x + 3 \)[/tex]. The slope ([tex]\( m \)[/tex]) of this line is [tex]\( \frac{2}{3} \)[/tex].
2. Use the x-intercept to find the y-intercept of the new line:
You are given that the x-intercept of the new line is -37. The x-intercept is the point where the line crosses the x-axis, that is, ([tex]\(-37, 0\)[/tex]).
3. Apply the point-slope form to find the y-intercept [tex]\( b \)[/tex]:
The equation of any line can be written as [tex]\( y = mx + b \)[/tex]. Plugging in the slope [tex]\( m = \frac{2}{3} \)[/tex] and the x-intercept point ([tex]\(-37, 0\)[/tex]) into this equation will allow us to solve for [tex]\( b \)[/tex].
[tex]\[
0 = \frac{2}{3}(-37) + b
\][/tex]
Calculate [tex]\( \frac{2}{3}(-37) = -24.666666666666664 \)[/tex].
So we have:
[tex]\[
0 = -24.666666666666664 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 24.666666666666664
\][/tex]
4. Write the final equation:
Now that we have both the slope [tex]\( m = \frac{2}{3} \)[/tex] and the y-intercept [tex]\( b = 24.666666666666664 \)[/tex], you can write the equation of the line:
[tex]\[
y = \frac{2}{3}x - 24.666666666666664
\][/tex]
So the equation of the line that is parallel to the given line and has an x-intercept of -37 is:
[tex]\[
y = \frac{2}{3}x - 24.666666666666664
\][/tex]
Hence, none of the provided multiple choices matches the exact line equation we've found. However, the correct line equation remains as derived above.
