Question:In the xy-plane, the graph of the equation 5x + 8y = 12 is translated 3 units to the right....
GMAT Algebra : (Alg) Questions
In the xy-plane, the graph of the equation \(5\mathrm{x} + 8\mathrm{y} = 12\) is translated \(3\) units to the right. What is the y-coordinate of the y-intercept of the resulting graph?
1. TRANSLATE the transformation information
- Given: Original equation \(5\mathrm{x} + 8\mathrm{y} = 12\) translated 3 units to the right
- Translation rule: To move right by 3 units, replace x with \(\mathrm{x} - 3\)
2. SIMPLIFY to find the new equation
- Substitute \(\mathrm{x} - 3\) for x in the original equation:
\(5(\mathrm{x} - 3) + 8\mathrm{y} = 12\)
- Expand:
\(5\mathrm{x} - 15 + 8\mathrm{y} = 12\)
- Rearrange:
\(5\mathrm{x} + 8\mathrm{y} = 27\)
3. INFER how to find the y-intercept
- The y-intercept occurs when \(\mathrm{x} = 0\)
- Substitute \(\mathrm{x} = 0\) into the transformed equation
4. SIMPLIFY to solve for y
\(5(0) + 8\mathrm{y} = 27\)
\(8\mathrm{y} = 27\)
\(\mathrm{y} = \frac{27}{8}\)
Answer: \(\frac{27}{8}\) (which equals 3.375)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the direction of horizontal translation and use \(\mathrm{x} + 3\) instead of \(\mathrm{x} - 3\) when translating "3 units to the right."
This leads them to the equation \(5(\mathrm{x} + 3) + 8\mathrm{y} = 12\), which simplifies to \(5\mathrm{x} + 8\mathrm{y} = -3\). Finding the y-intercept gives \(\mathrm{y} = -\frac{3}{8}\), leading to confusion since this likely doesn't match any reasonable answer choice, causing them to guess.
Second Most Common Error:
Poor INFER reasoning: Students find the y-intercept of the original equation \((5\mathrm{x} + 8\mathrm{y} = 12)\) instead of the transformed equation, getting \(\mathrm{y} = \frac{12}{8} = \frac{3}{2} = 1.5\).
This happens because they don't recognize that the transformation changes the equation itself, and they need to work with the new equation to find the new y-intercept.
The Bottom Line:
This problem tests whether students truly understand that transforming a graph means transforming its equation, and that they must work systematically through the transformation before finding specific features like intercepts.