Problem 89 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The graphs of \(2 x-3 y=-18\) and \(-2 x+3 y=18\) must have the same intercepts because I can see that the equations are equivalent.

Step-by-Step Solution

Answer

The statement makes sense. The two equations are equivalent, so their graphs are identical, thus must have the same x and y intercepts.

Step 1: Investigate Equation Equivalence

Examine the two equations \(2x - 3y = -18\) and \(-2x + 3y = 18\). Transform the second equation by multiplying it with -1. After multiplying, the equation becomes \(2x - 3y = -18\), which is identical to the first equation, so they are indeed equivalent.

Step 2: Intercept Calculation

Calculate the x and y intercepts. The x-intercept is calculated by setting y = 0 in the equation and solving for x. The y-intercept is calculated by setting x = 0 in the equation and solving for y.

Step 3: Intercept Comparison

Compare the intercepts of the two equations. Since the equations are equivalent, they will have the identical intercepts. Hence, the statement indeed makes sense.