To find the value of [tex]\( x \)[/tex] that satisfies the given conditions, we follow these steps:
1. Calculate the slope of the first line passing through points (8, 7) and (7, -6):
The slope formula is given by:
[tex]\[
\text{slope}_1 = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((8, 7)\)[/tex] and [tex]\((7, -6)\)[/tex]:
[tex]\[
\text{slope}_1 = \frac{-6 - 7}{7 - 8} = \frac{-13}{-1} = 13
\][/tex]
2. Determine the slope of the second line that is perpendicular to the first line:
If two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex]. Let the slope of the second line be [tex]\(\text{slope}_2\)[/tex]. Thus:
[tex]\[
\text{slope}_1 \cdot \text{slope}_2 = -1
\][/tex]
Substituting the value of [tex]\(\text{slope}_1\)[/tex]:
[tex]\[
13 \cdot \text{slope}_2 = -1
\][/tex]
Solving for [tex]\(\text{slope}_2\)[/tex]:
[tex]\[
\text{slope}_2 = -\frac{1}{13}
\][/tex]
3. Use the slope of the second line to find [tex]\( x \)[/tex] for the line passing through points (2, 4) and [tex]\((x, 3)\)[/tex]:
The slope formula for the second line is:
[tex]\[
\text{slope}_2 = \frac{y_4 - y_3}{x_4 - x_3}
\][/tex]
Substituting the points [tex]\((2, 4)\)[/tex] and [tex]\((x, 3)\)[/tex] and the value of [tex]\(\text{slope}_2\)[/tex]:
[tex]\[
-\frac{1}{13} = \frac{3 - 4}{x - 2}
\][/tex]
Simplify the numerator:
[tex]\[
-\frac{1}{13} = \frac{-1}{x - 2}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[
-1 \cdot (x - 2) = -13 \cdot 1
\][/tex]
Simplify:
[tex]\[
-x + 2 = -13
\][/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[
-x = -13 - 2
\][/tex]
[tex]\[
-x = -15
\][/tex]
[tex]\[
x = 15
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the given conditions is [tex]\(\boxed{15}\)[/tex].
