To find the value of [tex]\( x \)[/tex] where the line passing through the points [tex]\((1, 2)\)[/tex] and [tex]\((x, 5)\)[/tex] is perpendicular to a line with a slope of [tex]\(\frac{1}{3}\)[/tex], follow these steps:
1. Determine the slope of the perpendicular line:
If a line has a slope of [tex]\(\frac{1}{3}\)[/tex], the slope of a line perpendicular to this line will be the negative reciprocal. The reciprocal of [tex]\(\frac{1}{3}\)[/tex] is [tex]\(3\)[/tex], so the perpendicular slope is [tex]\(-3\)[/tex].
2. Set up the equation for the slope of the line passing through [tex]\((1, 2)\)[/tex] and [tex]\((x, 5)\)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given the points [tex]\((1, 2)\)[/tex] and [tex]\((x, 5)\)[/tex], we substitute these coordinates into the formula:
[tex]\[
\text{slope} = \frac{5 - 2}{x - 1} = \frac{3}{x - 1}
\][/tex]
3. Set the calculated slope equal to the perpendicular slope:
Since the slope of the line passing through [tex]\((1, 2)\)[/tex] and [tex]\((x, 5)\)[/tex] must equal the slope of the perpendicular line [tex]\(-3\)[/tex], we write the equation:
[tex]\[
\frac{3}{x - 1} = -3
\][/tex]
4. Solve the equation for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], start by multiplying both sides of the equation by [tex]\( x - 1 \)[/tex] to clear the fraction:
[tex]\[
3 = -3(x - 1)
\][/tex]
Distribute the [tex]\(-3\)[/tex] on the right side:
[tex]\[
3 = -3x + 3
\][/tex]
Move the constants to one side:
[tex]\[
3 - 3 = -3x
\][/tex]
Simplifying, you get:
[tex]\[
0 = -3x
\][/tex]
Divide both sides by [tex]\(-3\)[/tex]:
[tex]\[
x = 0
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{0} \)[/tex].
