Problem 24 - College Algebra

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 line is \(y = -1/4x - 6\).

Step 1: Find the slope of the given line

First, reformat the given equation \(4x - y - 6 = 0\) to slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Add \(y\) and subtract \(4x\) from both sides of the equation to get \(y = 4x - 6\). From this, we can see that the slope of the given line \(m = 4.\)

Step 2: Find the slope of the line to be found

Since the line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of the slope of the given line. Thus, the slope of the line to be found is \(-1/4.\)

Step 3: Find the y-intercept of the line to be found

The problem states that the line we are looking for has the same y-intercept as the given line. From step 1, we found the y-intercept of the given line to be \(-6.\) So, the y-intercept of the line we are looking for is also \(-6.\)

Step 4: Write the equation of the line to be found

Now we have the slope and y-intercept for the line we are trying to find. Insert these values into the slope-intercept formula to get the equation of the line. It is \(y = -1/4x - 6\).