Questions: Given the line -8x + 4y = -24. Determine the equation of the perpendicular and parallel line through the point (12,8). (a) parallel line

Transcript text: 2. Given the line $-8 x+4 y=-24$. Determine the equation of the perpendicular and parallel line through the point $(12,8)$. (a) parallel line

Solution

Solution Steps

To find the equation of a line parallel to the given line, we need to use the same slope. First, convert the given line to slope-intercept form to find its slope. Then, use the point-slope form of a line equation with the given point to find the equation of the parallel line.

Step 1: Find the Slope of the Given Line

The equation of the given line is

\[ -8x + 4y = -24. \]

To find the slope, we convert this equation to slope-intercept form $y = mx + b$:

\[ 4y = 8x - 24 \implies y = 2x - 6. \]

Thus, the slope $m$ of the given line is $2$.

Step 2: Use the Point-Slope Form for the Parallel Line

The equation of a line parallel to the given line will have the same slope $m = 2$ and pass through the point $(12, 8)$. We use the point-slope form of the line equation:

\[ y - y_1 = m(x - x_1), \]

where $(x_1, y_1) = (12, 8)$:

\[ y - 8 = 2(x - 12). \]

Step 3: Simplify the Equation

Expanding the equation gives:

\[ y - 8 = 2x - 24. \]

Adding $8$ to both sides results in:

\[ y = 2x - 16. \]