Question

y = 3x – 1
COLUMN A: x
COLUMN B: \(\frac{y}{3}+3\)

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

When a question provides an equation relating two variables, it's almost always a good idea to use substitution to express both columns in terms of a single variable. This makes the comparison direct and avoids the need for test cases.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are given an equation relating \(x\) and \(y\). We need to compare \(x\) with an expression involving \(y\). The best strategy is to express both columns in terms of the same variable.
Step 2: Key Formula or Approach:
Substitute the given expression for \(y\) into Column B to express Column B in terms of \(x\). Then, compare the resulting expression with \(x\).
Step 3: Detailed Explanation:
Column A: The quantity is \(x\). Column B: The quantity is \( \frac{y}{3} + 3 \). We are given that \( y = 3x - 1 \). Let's substitute this into the expression for Column B. \[ \text{Column B} = \frac{(3x - 1)}{3} + 3 \] Split the fraction: \[ \text{Column B} = \frac{3x}{3} - \frac{1}{3} + 3 \] \[ \text{Column B} = x - \frac{1}{3} + 3 \] Combine the constant terms: \[ \text{Column B} = x + \left(3 - \frac{1}{3}\right) = x + \frac{9}{3} - \frac{1}{3} = x + \frac{8}{3} \] So, the quantity in Column B is \( x + \frac{8}{3} \). Comparison: We are comparing \(x\) (Column A) with \(x + \frac{8}{3}\) (Column B). Since \( \frac{8}{3} \) is a positive number (\( \frac{8}{3} \approx 2.67 \)), the expression \(x + \frac{8}{3}\) will always be greater than \(x\). Therefore, the quantity in Column B is greater.
Step 4: Final Answer:
By expressing Column B in terms of \(x\), we find it is equal to \(x + \frac{8}{3}\), which is always greater than \(x\).