To find the equation of a line that is parallel to the given line [tex]\( y = \frac{2}{3}x + 3 \)[/tex] and has an [tex]\( x \)[/tex]-intercept of [tex]\(-3\)[/tex], follow these steps:
1. Determine the slope of the given line:
The slope ([tex]\( m \)[/tex]) of the given line [tex]\( y = \frac{2}{3}x + 3 \)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
2. Identify the slope of the parallel line:
Since parallel lines have the same slope, the slope of our desired line is also [tex]\(\frac{2}{3}\)[/tex].
3. Use the [tex]\( x \)[/tex]-intercept to find the y-intercept ([tex]\( b \)[/tex]):
An [tex]\( x \)[/tex]-intercept at [tex]\(-3\)[/tex] means that the line passes through the point [tex]\((-3,0)\)[/tex]. We use this point to find the y-intercept of the new line.
The equation of the line we are looking for is of the form [tex]\( y = \frac{2}{3}x + b \)[/tex].
Substitute the point [tex]\((-3,0)\)[/tex] into the equation:
[tex]\[
0 = \frac{2}{3}(-3) + b
\][/tex]
4. Solve for [tex]\( b \)[/tex]:
Calculate:
[tex]\[
0 = -2 + b
\][/tex]
Add 2 to both sides:
[tex]\[
b = 2
\][/tex]
5. Write the equation of the line:
Now that we have the slope [tex]\( \frac{2}{3} \)[/tex] and the y-intercept [tex]\( 2 \)[/tex], we can write the equation of the line:
[tex]\[
y = \frac{2}{3}x + 2
\][/tex]
Among the given options, this matches:
[tex]\[
y = \frac{2}{3}x + 2
\][/tex]
So, the equation of the line that is parallel to the given line and has an [tex]\( x \)[/tex]-intercept of [tex]\(-3\)[/tex] is:
[tex]\[
y = \frac{2}{3}x + 2
\][/tex]
