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Difficulty: EasyAlgebraic Equations and Systems in Data Sufficiency

If aa and bb are real numbers, what is the value of a2−b2a^2 - b^2?

(1) a+b=7a + b = 7
(2) a−b=3a - b = 3

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
Factoring the target expression yields a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b). Neither statement alone gives both factors, so neither statement alone is sufficient. Combining Statement (1), which gives a+b=7a + b = 7, and Statement (2), which gives a−b=3a - b = 3, allows us to calculate a2−b2=7×3=21a^2 - b^2 = 7 \times 3 = 21. Therefore, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
Recognize that a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b). The objective is to find the value of the product (a+b)(a−b)(a + b)(a - b).
Simplifying the target expression reduces the problem to finding the sum and difference of the variables.
2
Evaluate Statement (1) independently.
Statement (1) states a+b=7a + b = 7. Knowing only a+ba + b leaves a−ba - b unknown, so a2−b2=7(a−b)a^2 - b^2 = 7(a - b) can take infinitely many values. Statement (1) is insufficient.
A single linear equation with two variables cannot determine the value of a two-factor product unless the second factor is fixed.
3
Evaluate Statement (2) independently.
Statement (2) states a−b=3a - b = 3. Knowing only a−ba - b leaves a+ba + b unknown, so a2−b2=3(a+b)a^2 - b^2 = 3(a + b) can take infinitely many values. Statement (2) is insufficient.
A single difference equation does not uniquely determine the sum of two variables.
4
Evaluate Statements (1) and (2) together.
Combining both statements gives a+b=7a + b = 7 and a−b=3a - b = 3. Substituting these values into the rephrased target expression yields a2−b2=(7)(3)=21a^2 - b^2 = (7)(3) = 21. A unique numerical value is obtained. Statements combined are sufficient.
Having both component factors allows direct computation of their product without needing to solve for individual values of aa and bb first.

Key Concept

Algebraic Rephrasing and Difference of Squares in Data Sufficiency
Estimated Time:1m 0s
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Topics

Algebraic Equations and Systems in Data Sufficiency

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