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Which equation represents a line parallel to [tex]\( 3x - 8y = 12 \)[/tex]?

A. [tex]\( y = \frac{3}{8}x - 4 \)[/tex]
B. [tex]\( y = -\frac{3}{8}x - 4 \)[/tex]
C. [tex]\( y = \frac{8}{3}x - 4 \)[/tex]
D. [tex]\( y = -\frac{8}{3}x - 4 \)[/tex]


Sagot :

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]
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