Problem 23 - 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 \(3 x-2 y-4=0\) and has the same \(y\) -intercept as this line.

Step-by-Step Solution

Answer

The equation of the line that is perpendicular to \(3x - 2y - 4 = 0\) and possess the same y-intercept is given by \(y = -\frac{2}{3}x + 2\)

Step 1: Find the Slope of the Original Line

First rearrange the given equation \(3 x-2 y-4=0\) to slope-intercept form \(y=mx+b\). The slope m of the original line can then be identified as the coefficient of x.

Step 2: Find the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the latter. Therefore, calculate the negative reciprocal of the slope found in step 1. This will be the slope m of the required line.

Step 3: Determine the y-intercept

The y-intercept is same for both lines as per the problem's conditions. So, identify the y-intercept b from the re-arranged given equation.

Step 4: Substitute Slope and y-intercept into Equation

Finally, substitute the slope and y-intercept obtained in steps 2 and 3 respectively into the slope-intercept form \(y = mx + b\). This will yield the equation of the required line.