Question

The line passing through the point
S (1,2) and (x,5) is perpendicular to a line that has a slope of 1/3 what is the value of x

Answer 1

To find the value of \( x \) such that the line passing through the points \( S(1,2) \) and \( (x, 5) \) is perpendicular to a line with a slope of \( \frac{1}{3} \), we can follow these steps:

1. **Identify the slope of the line through points \( S(1,2) \) and \( (x, 5) \)**:
The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For our points, this becomes:
\[
m = \frac{5 - 2}{x - 1} = \frac{3}{x - 1}
\]

2. **Determine the slope of the line that is perpendicular to the given line**:
The slope of the given line is \( \frac{1}{3} \). The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Thus, the slope we want is:
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{3}} = -3
\]

3. **Set the slope we calculated equal to the slope of the perpendicular line**:
We have:
\[
\frac{3}{x - 1} = -3
\]

4. **Solve for \( x \)**:
\[
3 = -3(x - 1)
\]
\[
3 = -3x + 3
\]
Subtract \( 3 \) from both sides:
\[
0 = -3x
\]
Dividing both sides by \( -3 \):
\[
x = 0
\]

Thus, the value of \( x \) is \( \boxed{0} \).