Problem 97 - College Algebra Essentials

Question

Explain how to use the general form of a line's equation to find the line's slope and \(y\) -intercept.

Step-by-Step Solution

Answer

One can find the slope and y-intercept by rearranging the general form \(Ax + By = C\) into slope-intercept form \(y = \frac{C}{B} - \frac{A}{B}x\). The slope will be \(-\frac{A}{B}\) and the y-intercept will be \(\frac{C}{B}\).

Step 1: Rearrange equation

The first step is to change the general form of the line's equation into the slope-intercept form. To do that, start from the general form of the line's equation, which is \(Ax + By = C\). Rearrange it to find \(y\) in terms of \(x\), which will yield the following: \(y = \frac{C}{B} - \frac{A}{B}x\).

Step 2: Identify slope

Now that the equation of the line is in slope-intercept form, the coefficient of \(x\) in the equation, which is \(-\frac{A}{B}\), is the slope of the line.

Step 3: Identify y-intercept

In the slope-intercept form of the line's equation, the constant term is the y-intercept, i.e., the value of \(y\) when \(x = 0\). So, \(\frac{C}{B}\) is the y-intercept of the line in this case.