What is the equation of the line that is parallel to the given line and has an [tex]\( x \)[/tex]-intercept of -3?

A. [tex]\( y = \frac{2}{3}x + 3 \)[/tex]
B. [tex]\( y = \frac{2}{3}x + 2 \)[/tex]
C. [tex]\( y = -\frac{3}{2}x + 3 \)[/tex]
D. [tex]\( y = \frac{3}{2}x + 2 \)[/tex]



Answer :

To find the equation of a line that is parallel to a given line and has an [tex]\( x \)[/tex]-intercept of [tex]\(-3\)[/tex], we need to start by identifying the slope of the given lines.

1. We are given the following lines:
[tex]\[ y = \frac{2}{3}x + 3, \][/tex]
[tex]\[ y = \frac{2}{3}x + 2, \][/tex]
[tex]\[ y = -\frac{3}{2}x + 3, \][/tex]
[tex]\[ y = \frac{3}{2}x + 2. \][/tex]

2. Observing these equations, we see that the first and second lines have a slope of [tex]\(\frac{2}{3}\)[/tex]. The third line has a slope of [tex]\(-\frac{3}{2}\)[/tex], and the fourth line has a slope of [tex]\(\frac{3}{2}\)[/tex].

3. Since the new line must be parallel to one of the given lines, it must have the same slope as one of them. The valid choices are the lines with a slope of [tex]\(\frac{2}{3}\)[/tex].

4. The slope of the new line will be [tex]\(\frac{2}{3}\)[/tex].

5. The next step is to use the [tex]\( x \)[/tex]-intercept of [tex]\(-3\)[/tex]. In the Slope-Intercept form of a line, [tex]\( y = mx + b \)[/tex], the coordinates of the [tex]\( x \)[/tex]-intercept are [tex]\((-3, 0)\)[/tex].

6. Plugging the point [tex]\((-3, 0)\)[/tex] into the equation to find the intercept [tex]\( b \)[/tex]:

[tex]\[ 0 = \frac{2}{3}(-3) + b. \][/tex]

7. Solving for [tex]\( b \)[/tex]:

[tex]\[ 0 = -2 + b \implies b = 2. \][/tex]

Thus, the equation of the new 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]