Question

two perpendicular lines intersect (meet) on the x - axis. if the first line is defined by the equation $y = \frac{1}{3}x - 2$, what is the equation of the other line?

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{1}{3}x - 2 \) is \( m_1=\frac{1}{3} \). For two perpendicular lines, the product of their slopes is \( - 1 \), i.e., \( m_1\times m_2=-1 \). Let the slope of the perpendicular line be \( m_2 \). Then \( \frac{1}{3}\times m_2=-1 \), so \( m_2=-3 \).

Step2: Find the intersection point on the x - axis

To find the intersection point of the given line with the x - axis, we set \( y = 0 \) in the equation \( y=\frac{1}{3}x - 2 \). So \( 0=\frac{1}{3}x-2 \). Solving for \( x \), we add 2 to both sides: \( \frac{1}{3}x = 2 \), then multiply both sides by 3: \( x = 6 \). So the intersection point is \( (6,0) \).

Step3: Use point - slope form to find the equation of the line

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(6,0) \) and \( m = - 3 \). Substituting these values, we get \( y-0=-3(x - 6) \). Simplifying this, we have \( y=-3x + 18 \).

Answer:

\( y=-3x + 18 \)