y = 18 - 5x The equation above represents the speed y, in feet per second, of Sheila's bicycle x...
GMAT Algebra : (Alg) Questions
\(\mathrm{y = 18 - 5x}\)
The equation above represents the speed \(\mathrm{y}\), in feet per second, of Sheila's bicycle \(\mathrm{x}\) seconds after she applied the brakes at the end of a ride. If the equation is graphed in the \(\mathrm{xy}\)-plane, which of the following is the best interpretation of the \(\mathrm{x}\)-coordinate of the line's \(\mathrm{x}\)-intercept in the context of the problem?
The speed of Sheila's bicycle, in feet per second, before Sheila applied the brakes
The number of feet per second the speed of Sheila's bicycle decreased each second after Sheila applied the brakes
The number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop
The number of feet Sheila's bicycle traveled from the time she began applying the brakes until the bicycle came to a complete stop
1. TRANSLATE the problem setup
- Given information:
- \(\mathrm{y = 18 - 5x}\) where \(\mathrm{y}\) = speed (ft/sec), \(\mathrm{x}\) = seconds after braking
- Need to interpret x-coordinate of x-intercept in context
- What this tells us: We need to find where the line crosses the x-axis and understand what that x-value means for Sheila's bicycle
2. TRANSLATE what "x-intercept" means mathematically
- The x-intercept occurs when \(\mathrm{y = 0}\)
- At this point, the bicycle's speed is 0 ft/sec (complete stop)
3. Find the x-intercept by setting y = 0
- \(\mathrm{0 = 18 - 5x}\)
- \(\mathrm{5x = 18}\)
- \(\mathrm{x = \frac{18}{5} = 3.6}\)
4. INFER the contextual meaning
- The x-intercept is at point \(\mathrm{(3.6, 0)}\)
- Since x represents "seconds after braking" and at this point \(\mathrm{y = 0}\) (stopped)
- The x-coordinate (3.6) represents the time it took from start of braking until complete stop
5. Match with answer choices
- Looking for "number of seconds until complete stop"
- This matches Choice C exactly
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse x-intercept with y-intercept or misinterpret what the coordinates represent in context.
They might think the x-intercept's x-coordinate represents the initial speed (confusing it with the y-intercept at \(\mathrm{x = 0}\)) or think it represents the rate of change (confusing coordinates with slope). This conceptual mix-up about intercepts and their contextual meaning leads to selecting incorrect interpretations.
This may lead them to select Choice A (initial speed) or Choice B (rate of decrease).
Second Most Common Error:
Inadequate INFER reasoning: Students find the x-intercept correctly but fail to connect that \(\mathrm{y = 0}\) means "complete stop" in the bicycle context.
They might calculate \(\mathrm{x = 3.6}\) but not realize this represents the moment when speed reaches zero. Without this connection, they can't properly interpret what this time value means for Sheila's bicycle journey.
This leads to confusion and guessing among the answer choices.
The Bottom Line:
This problem requires students to bridge pure mathematical concepts (intercepts, coordinates) with real-world interpretation (bicycle braking). The key insight is recognizing that mathematical events (\(\mathrm{y = 0}\)) correspond to meaningful physical events (complete stop).
The speed of Sheila's bicycle, in feet per second, before Sheila applied the brakes
The number of feet per second the speed of Sheila's bicycle decreased each second after Sheila applied the brakes
The number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop
The number of feet Sheila's bicycle traveled from the time she began applying the brakes until the bicycle came to a complete stop