Question: The graph of 7x + 5y = 12 is translated left 3 units in the xy-plane. What is the...
GMAT Algebra : (Alg) Questions
Question: The graph of \(7\mathrm{x} + 5\mathrm{y} = 12\) is translated left 3 units in the \(\mathrm{xy}\)-plane. What is the \(\mathrm{y}\)-coordinate of the \(\mathrm{y}\)-intercept of the resulting graph?
Express your answer as a fraction in lowest terms.
Format: Fill-in-the-blank (grid-in)
1. TRANSLATE the transformation information
- Given information:
- Original equation: \(7\mathrm{x} + 5\mathrm{y} = 12\)
- Translation: left 3 units
- What this tells us: A horizontal translation left by 3 units means replacing x with \(\mathrm{x} + 3\) in the equation
2. SIMPLIFY the transformed equation
- Apply the translation: \(7(\mathrm{x} + 3) + 5\mathrm{y} = 12\)
- Expand using distributive property: \(7\mathrm{x} + 21 + 5\mathrm{y} = 12\)
- Rearrange to standard form: \(7\mathrm{x} + 5\mathrm{y} = 12 - 21\)
- Final simplified form: \(7\mathrm{x} + 5\mathrm{y} = -9\)
3. INFER how to find the y-intercept
- The y-intercept occurs where the graph crosses the y-axis
- This happens when \(\mathrm{x} = 0\)
- Substitute \(\mathrm{x} = 0\) into our equation: \(7(0) + 5\mathrm{y} = -9\)
4. SIMPLIFY to find the final answer
- Solve for y: \(5\mathrm{y} = -9\)
- Therefore: \(\mathrm{y} = -\frac{9}{5}\)
Answer: \(-\frac{9}{5}\)
Alternative acceptable answers: \(-1.8\), \(-1\frac{4}{5}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students often confuse the direction of translation, thinking "left 3 units" means replacing x with \(\mathrm{x} - 3\) instead of \(\mathrm{x} + 3\).
This leads them to work with \(7(\mathrm{x} - 3) + 5\mathrm{y} = 12\), which simplifies to \(7\mathrm{x} + 5\mathrm{y} = 33\). Setting \(\mathrm{x} = 0\) gives \(\mathrm{y} = \frac{33}{5}\), a completely different answer that leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(7(\mathrm{x} + 3) + 5\mathrm{y} = 12\) but make algebraic errors during expansion or rearrangement.
Common mistakes include forgetting to distribute the 7 to both terms, or incorrectly combining constants (21 and 12). These algebraic slips lead to wrong equations and incorrect y-intercepts.
The Bottom Line:
The key challenge is correctly translating the English description of the transformation into proper mathematical notation. Once that's done correctly, the rest is straightforward algebra.