Problem 110 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.

Step-by-Step Solution

Answer

The statement is false. Not all lines in the rectangular coordinate system can be expressed in the slope-intercept form, such as vertical lines which have undefined slopes.

Step 1: Understanding the statement

The given statement is 'Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.' We need to determine if this is true or false.

Step 2: Evaluating the statement

The slope-intercept form is \(y = mx + b\), where m represents slope and b the y-intercept. This form can represent any line so far the line is not vertical. A vertical line, parallel to y-axis, hasn't a defined slope and therefore cannot be expressed in slope-intercept form.

Step 3: Correcting the statement (if necessary)

Since the statement is false, it needs to be corrected to be true. We could say instead: 'Most lines in the rectangular coordinate system can have equations that are expressed in slope-intercept form, with the exception of vertical lines, because they have an undefined slope.'