To determine the equation of a line that is parallel to a given line and passes through a specific point (in this case, having an x-intercept of -3), we need to follow several steps.
### Step 1: Identify the slope of the given line
We know that parallel lines have the same slope. Our goal is to find the slope of the given line, which in this case is of the form [tex]\( y = mx + b \)[/tex].
Given equations:
1. [tex]\( y = \frac{2}{3} x + 3 \)[/tex]
2. [tex]\( y = \frac{2}{3} x + 2 \)[/tex]
3. [tex]\( y = -\frac{3}{2} x + 3 \)[/tex]
4. [tex]\( y = -\frac{3}{2} x + 2 \)[/tex]
All these equations are in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We are looking for the slope, [tex]\( m \)[/tex], for the lines that are parallel.
### Step 2: Select the slope for the parallel line
The slope of the lines that are parallel must be the same as in the given options. Therefore, the slope, [tex]\( m \)[/tex], will be [tex]\( \frac{2}{3} \)[/tex] from the lines [tex]\( y = \frac{2}{3} x + 3 \)[/tex] and [tex]\( y = \frac{2}{3} x + 2 \)[/tex].
### Step 3: Determine the y-intercept ([tex]\( b \)[/tex]) for the required line
The line should pass through the x-intercept of -3. For the x-intercept -3, this means that when [tex]\( x = -3 \)[/tex], [tex]\( y = 0 \)[/tex].
We can use the point-slope form of the equation [tex]\( y = mx + b \)[/tex] and solve for [tex]\( b \)[/tex]:
[tex]\[
0 = \left(\frac{2}{3}\right)(-3) + b
\][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[
0 = -2 + b \implies b = 2
\][/tex]
### Step 4: Form the equation of the line
Now that we have the slope [tex]\( \frac{2}{3} \)[/tex] and the y-intercept [tex]\( b = 2 \)[/tex], we can write the equation of the line as:
[tex]\[
y = \frac{2}{3} x + 2
\][/tex]
### Final Answer:
The equation of the line that is parallel to the given lines and has an x-intercept of -3 is:
[tex]\[
y = \frac{2}{3} x + 2
\][/tex]
