Question

the slope of the line passing through points $(x_1,y_1)$ and $(x_2,y_2)$ is found using the formula $\frac{y_2 - y_1}{x_2 - x_1}$. the line passing through the points $(1,2)$ and $(x,5)$ is perpendicular to a line that has a slope of $\frac{1}{3}$. what is the value of $x$? a 10 b -2 c -4 d 0

Explanation:

Step1: Find the slope of the line passing through (1, 2) and (x, 5)

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $y_2 = 5$, $y_1=2$, $x_2=x$, $x_1 = 1$, so the slope $m_1=\frac{5 - 2}{x - 1}=\frac{3}{x - 1}$.

Step2: Use the perpendicular - slope relationship

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the other line be $m_2=\frac{1}{3}$. Then $m_1\times m_2=-1$. Substitute $m_1$ and $m_2$ into the equation: $\frac{3}{x - 1}\times\frac{1}{3}=-1$.

Step3: Solve the equation for x

Simplify the left - hand side of the equation: $\frac{3}{3(x - 1)}=\frac{1}{x - 1}$. So, $\frac{1}{x - 1}=-1$. Cross - multiply to get $1=-(x - 1)$. Expand the right - hand side: $1=-x + 1$. Subtract 1 from both sides: $0=-x$. Then $x = 0$.

Answer:

D. 0