To solve this problem, we need to first identify the slope of the given line [tex]\(3x - 8y = 12\)[/tex] and then find the equation that has the same slope, as parallel lines have the same slope.
1. Convert the given line's equation [tex]\(3x - 8y = 12\)[/tex] to the slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Start by isolating [tex]\(y\)[/tex] on one side of the equation:
[tex]\[
3x - 8y = 12
\][/tex]
- Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
-8y = -3x + 12
\][/tex]
- Divide every term by [tex]\(-8\)[/tex] to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{3}{8}x - \frac{12}{8}
\][/tex]
- Simplify the constant term:
[tex]\[
y = \frac{3}{8}x - \frac{3}{2}
\][/tex]
Now, the equation is in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
2. Identify the slope of the given line:
The slope [tex]\(m\)[/tex] from the slope-intercept form [tex]\(y = \frac{3}{8}x - \frac{3}{2}\)[/tex] is:
[tex]\[
m = \frac{3}{8}
\][/tex]
3. Examine the options to find which equation has the same slope [tex]\(\frac{3}{8}\)[/tex]:
- Option A: [tex]\(y = \frac{3}{8}x - 4\)[/tex]
- The slope is [tex]\(\frac{3}{8}\)[/tex], which matches the slope of the given line.
- Option B: [tex]\(y = -\frac{3}{8}x - 4\)[/tex]
- The slope is [tex]\(-\frac{3}{8}\)[/tex], which does not match the given slope.
- Option C: [tex]\(y = \frac{8}{3}x - 4\)[/tex]
- The slope is [tex]\(\frac{8}{3}\)[/tex], which does not match the given slope.
- Option D: [tex]\(y = -\frac{8}{3}x - 4\)[/tex]
- The slope is [tex]\(-\frac{8}{3}\)[/tex], which does not match the given slope.
4. Conclusion:
The only option with the same slope [tex]\(\frac{3}{8}\)[/tex] as the given line is [tex]\(\boxed{A}\)[/tex].
Thus, the correct equation that represents a line parallel to [tex]\(3x - 8y = 12\)[/tex] is:
[tex]\[
\boxed{y = \frac{3}{8}x - 4}
\][/tex]
