Problem 24 - College Algebra Essentials

Question

Write an equation in slope-intercept form of a linear function \(f\) whose graph satisfies the given conditions. The graph of \(f\) is perpendicular to the line whose equation is \(4 x-y-6=0\) and has the same \(y\) -intercept as this line.

Step-by-Step Solution

Answer

The equation of the required line, in slope-intercept form, is \(y = -1/4x - 6\).

Step 1: Rearrange the given equation in slope-intercept form

To find the slope and y-intercept of the given line, the equation needs to be in the form \(y = mx + b\). So, rearrange the given equation to that form: \(4x - y = 6\) becomes \(y = 4x - 6\). From this, you can tell that the slope \(m\) of the original line is 4 and the y-intercept \(b\) is -6.

Step 2: Find the slope of the required line

The slope of the line we are looking to write the equation for is the negative reciprocal of the slope of the given line. That is, if the slope of the given line is 4, then the slope of the required line is \(-1/4\).

Step 3: Write the equation of the required line

The line we are interested in has the same y-intercept as the given line and a slope that is the negative reciprocal of the slope of the given line. From Steps 1 and 2, we know these values to be -6 and -1/4, respectively. Substituting these into the formula \(y = mx + b\), the equation of the required line is \(y = -1/4x - 6\).