Problem 51 - Introductory and Intermediate Algebra for College Students 4th

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 equation for the line passing through \((2,-3)\) and perpendicular to \(y=\frac{1}{5} x+6\) is \(y = -5x + 7\) in slope-intercept form

Step 1: Find the slope of the perpendicular line

The slope of the given line, represented by 'm' in the equation \(y = mx + c\), is \(1/5\). The slope, say \(m'\), of a line perpendicular to it would be the negative reciprocal of \(1/5\) . Thus, \(m' = - (1 / (1/5)) = -5\)

Step 2: Write the equation in point-slope form

Use the point-slope form of an equation, \(y - y1 = m'(x - x1)\), where \(x1 = 2, y1 = -3, m' = -5\). Substituting these values gives the point-slope form equation of the perpendicular line, which is \(y - (-3) = -5 (x - 2)\), or \(y + 3 = -5x + 10\)

Step 3: Convert the equation to slope-intercept form

Slope-intercept form of an equation is \(y = mx + c\). Converting the equation \(y + 3 = -5x + 10\) to slope-intercept form gives \(y = -5x + 7\).