To determine the slope of the line given by the equation [tex]\(3x - 4y = 7\)[/tex], we need to rearrange the equation into the slope-intercept form, which is [tex]\(y = mx + b\)[/tex]. In this form, [tex]\(m\)[/tex] represents the slope, and [tex]\(b\)[/tex] represents the y-intercept.
Here are the steps to convert the given equation into slope-intercept form:
1. Start with the equation of the line:
[tex]\[
3x - 4y = 7
\][/tex]
2. We need to isolate [tex]\(y\)[/tex]. First, move the term involving [tex]\(x\)[/tex] to the other side by subtracting [tex]\(3x\)[/tex] from both sides:
[tex]\[
-4y = -3x + 7
\][/tex]
3. Next, divide every term by [tex]\(-4\)[/tex] to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-3x + 7}{-4}
\][/tex]
4. Simplify the fraction by distributing the division:
[tex]\[
y = \frac{3}{4}x - \frac{7}{4}
\][/tex]
Now, the equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where the coefficient of [tex]\(x\)[/tex] (which is [tex]\(\frac{3}{4}\)[/tex]) represents the slope [tex]\(m\)[/tex].
Therefore, the slope of the line given by the equation [tex]\(3x - 4y = 7\)[/tex] is:
[tex]\(
\frac{3}{4}
\)[/tex]
So, the correct answer is:
(C) [tex]\(\frac{3}{4}\)[/tex]
