Problem 12 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through \((5,-9)\) and perpendicular to the line whose equation is \(x+7 y-12=0\)

Step-by-Step Solution

Answer

The equation in point-slope form is \(y+9=7(x-5)\) and the equation in general form is \(7x - y - 44 = 0\).

Step 1: Find the slope of the given line

Rewrite the given line equation in slope-intercept form, which is \(y = mx + b\). To do this, isolate \(y\) in the equation \(x+7y-12=0\) to get \(y = -1/7x + 12/7\). Thus, the slope of the line, \(m_1\), is -1/7.

Step 2: Find the slope of the required line

The line required is perpendicular to the given line. Therefore, its slope, \(m_2\), is the negative reciprocal of \(m_1\), which means \(m_2 = -1/-1/7 = 7\).

Step 3: Write the equation in point-slope form

Now substitute values of \(m_2\), \(x_1\) and \(y_1\) into the point-slope form equation. The equation becomes \(y+9=7(x-5)\).

Step 4: Convert the equation into general form

Expand and rearrange the equation. From \(y+9 = 7x -35\), you can rearrange it into \(7x - y -44 = 0\) which is the equation in general form.