To find the slope of a line that is perpendicular to the line given by the equation [tex]\(2x + 7y = 5\)[/tex], follow these steps:
1. Convert the given equation to slope-intercept form:
The slope-intercept form of a line's equation is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Starting with the given equation:
[tex]\[
2x + 7y = 5
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
7y = -2x + 5
\][/tex]
[tex]\[
y = \left( -\frac{2}{7} \right)x + \frac{5}{7}
\][/tex]
This equation is now in the form [tex]\(y = mx + b\)[/tex], where:
[tex]\[
m = -\frac{2}{7}
\][/tex]
So, the slope [tex]\(m\)[/tex] of the original line is [tex]\(-\frac{2}{7}\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of [tex]\(m\)[/tex] is found by flipping the fraction and changing the sign.
The original slope is:
[tex]\[
m = -\frac{2}{7}
\][/tex]
Therefore, the negative reciprocal is:
[tex]\[
\frac{7}{2}
\][/tex]
3. Select the correct answer choice:
Among the given options, we determine that:
[tex]\[
\frac{7}{2}
\][/tex]
matches the slope of the line perpendicular to the given line.
Thus, the correct answer is:
D. [tex]\(\frac{7}{2}\)[/tex]
