To find the equation of the line that is parallel to a given line and has a specified \(x\)-intercept, we can follow these steps:
1. Determine the slope of the given line: The slope of the linear equation \(y = mx + b\) is \(m\). For the given line equations:
[tex]\[
y = \frac{2}{3} x + 3 \quad \text{and} \quad y = \frac{2}{3} x + 2
\][/tex]
Both lines have the same slope, \(m = \frac{2}{3}\).
2. Understand what it means to be parallel: Lines that are parallel to each other have the same slope. Therefore, our parallel line must also have a slope of \(\frac{2}{3}\).
3. Utilize the \(x\)-intercept given: An \(x\)-intercept is a point on the line where \(y = 0\). Given that the \(x\)-intercept is \(-3\), this means the line passes through the point \((-3, 0)\).
4. Find the y-intercept \(b\): We can use the slope-intercept form of the equation of a line:
[tex]\[
y = mx + b
\][/tex]
We substitute \(m = \frac{2}{3}\) and \((-3, 0)\) into the equation to solve for \(b\):
[tex]\[
0 = \frac{2}{3}(-3) + b
\][/tex]
Solving this:
[tex]\[
0 = -2 + b
\][/tex]
[tex]\[
b = 2
\][/tex]
5. Write the equation of the parallel line: Now that we have the slope \(\frac{2}{3}\) and the y-intercept \(b = 2\), we can write the equation of the line:
[tex]\[
y = \frac{2}{3} x + 2
\][/tex]
Thus, the equation of the line that is parallel to the given line and has an \(x\)-intercept of \(-3\) is:
[tex]\[
\boxed{y = \frac{2}{3} x + 2}
\][/tex]
