Problem 63 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((2,-3)\) and perpendicular to the line whose equation is \(y=\frac{1}{5} x+6\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y+3 = -5x + 10\) and the slope-intercept form is \(y = -5x + 7\).

Step 1: Determine the Slope

The given line has the slope \(\frac{1}{5}\). The slope of a line perpendicular to this one is the negative reciprocal of this slope. Hence the slope \(m\) of the required line is \(-\frac{1}{ \frac{1}{5}} = -5.\)

Step 2: Find the Point-Slope Form

The point-slope form of the line equation is obtained by substituting the known point \((2, -3)\) and the newly found slope \(-5\) into the point-slope formula \(y - y_1 = m(x - x_1)\). This gives: \(y - (-3) = -5(x - 2)\), which simplifies to \(y+3 = -5x + 10\).

Step 3: Find the Slope-Intercept Form

To find the slope-intercept form \(y=mx+c\), rearrange the point-slope equation to solve for \(y\). Here, that yields \(y = -5x + 10 - 3\), which simplifies to \(y = -5x + 7\).