A line in the xy-plane decreases by 3 units in y for every increase of 1 unit in x. The...
GMAT Algebra : (Alg) Questions
A line in the \(\mathrm{xy}\)-plane decreases by 3 units in \(\mathrm{y}\) for every increase of 1 unit in \(\mathrm{x}\). The line crosses the \(\mathrm{y}\)-axis at 60. What is the \(\mathrm{y}\)-value on the line when \(\mathrm{x = -8}\)?
1. TRANSLATE the problem information
- Given information:
- "decreases by 3 units in y for every increase of 1 unit in x" → \(\mathrm{slope = -3}\)
- "crosses the y-axis at 60" → \(\mathrm{y\text{-}intercept = 60}\)
- Find y when \(\mathrm{x = -8}\)
2. INFER the approach
- Since we have slope and y-intercept, we can write the equation using slope-intercept form
- Then substitute the given x-value to find the corresponding y-value
3. Write the equation using slope-intercept form
- Using \(\mathrm{y = mx + b}\) where \(\mathrm{m = -3}\) and \(\mathrm{b = 60}\):
- \(\mathrm{y = -3x + 60}\)
4. SIMPLIFY by substituting \(\mathrm{x = -8}\)
- \(\mathrm{y = -3(-8) + 60}\)
- \(\mathrm{y = 24 + 60}\) [Remember: negative times negative equals positive]
- \(\mathrm{y = 84}\)
Answer: E (84)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "decreases by 3 units in y for every increase of 1 unit in x" and think the slope is +3 instead of -3.
They reason: "The line goes down 3, so the slope is 3" without recognizing that a decrease means the slope should be negative. Using \(\mathrm{y = 3x + 60}\) and substituting \(\mathrm{x = -8}\) gives:
\(\mathrm{y = 3(-8) + 60}\)
\(\mathrm{y = -24 + 60}\)
\(\mathrm{y = 36}\)
This leads them to select Choice A (36).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify the slope as -3 and y-intercept as 60, but make an arithmetic error with the double negative.
They compute \(\mathrm{-3(-8) = -24}\) instead of +24, leading to:
\(\mathrm{y = -24 + 60 = 36}\)
or they might add incorrectly and get:
\(\mathrm{y = 24 + 60 = 74}\)
(though this isn't an answer choice, leading to confusion and guessing).
The Bottom Line:
This problem tests whether students can accurately translate rate of change language into mathematical slope notation, particularly recognizing that "decreases" indicates a negative slope.