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Algebraic Equations and Systems in Data Sufficiency

42 questions

Question 41Question →

If xx and yy are real numbers, what is the value of (x+y)2(x + y)^2?

(1) x2+y2=25x^2 + y^2 = 25
(2) x2−y2=7x^2 - y^2 = 7

Show answer & explanation

Answer: Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
The correct response states that Statements (1) and (2) together are not sufficient because solving the system yields x2=16x^2 = 16 and y2=9y^2 = 9. This gives x=±4x = \pm 4 and y=±3y = \pm 3, allowing xyxy to be either 1212 or −12-12. Consequently, (x+y)2=x2+y2+2xy(x + y)^2 = x^2 + y^2 + 2xy can equal either 4949 or 11, which prevents finding a single unique value.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities
(x+y)2=x2+y2+2xy(x + y)^2 = x^2 + y^2 + 2xy
Expanding the square shows that we need the sum of squares x2+y2x^2 + y^2 and the product term 2xy2xy (or the individual values of xx and yy) to determine a unique value.
2
Evaluate Statement (1) alone
x2+y2=25x^2 + y^2 = 25, but xyxy can take multiple values.
For example, if x=5x = 5 and y=0y = 0, then (x+y)2=25(x + y)^2 = 25. If x=4x = 4 and y=3y = 3, then (x+y)2=49(x + y)^2 = 49. Multiple values are possible, so Statement (1) alone is insufficient.
3
Evaluate Statement (2) alone
x2−y2=7x^2 - y^2 = 7, but (x+y)2(x + y)^2 is not uniquely determined.
If x=4x = 4 and y=3y = 3, then x2−y2=16−9=7x^2 - y^2 = 16 - 9 = 7 and (x+y)2=49(x + y)^2 = 49. If x=7x = \sqrt{7} and y=0y = 0, then x2−y2=7x^2 - y^2 = 7 and (x+y)2=7(x + y)^2 = 7. Multiple values are possible, so Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined
x2=16x^2 = 16 and y2=9y^2 = 9, which leads to x=±4x = \pm 4 and y=±3y = \pm 3.
Adding the two equations gives 2x2=32  ⟹  x2=162x^2 = 32 \implies x^2 = 16. Subtracting gives 2y2=18  ⟹  y2=92y^2 = 18 \implies y^2 = 9. Thus x=4x = 4 or −4-4, and y=3y = 3 or −3-3.
5
Check for sign ambiguity in the cross-term xyxy
If x=4,y=3x = 4, y = 3, then (x+y)2=72=49(x + y)^2 = 7^2 = 49. If x=4,y=−3x = 4, y = -3, then (x+y)2=12=1(x + y)^2 = 1^2 = 1.
Because the system of non-linear equations determines x2x^2 and y2y^2 but not the relative signs of xx and yy, there are two distinct valid values for (x+y)2(x + y)^2. Therefore, Statements (1) and (2) together are NOT sufficient.

Key Concept

Non-linear systems of equations and degree ambiguity in Data Sufficiency
Estimated Time:2m 0s
Question 42Question →

If aa and bb are real numbers, what is the value of a3−b3a^3 - b^3?

(1) a−b=4a - b = 4
(2) a2+b2=26a^2 + b^2 = 26

Show answer & explanation

Answer: BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct answer choice states that both statements together are sufficient, but neither alone is sufficient. Statement (1) gives the linear difference between the variables, and Statement (2) gives the sum of their squares. Neither alone yields a single value for a3−b3a^3 - b^3. However, squaring Statement (1) allows us to isolate the product ab=5ab = 5. Using the standard factoring identity a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2), we can substitute the known components to find the unique value 124.

Step-by-Step Solution

1
Rephrase the target expression using algebraic factoring identities.
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Expanding or factoring the expression reveals that we need the values of (a−b)(a - b), (a2+b2)(a^2 + b^2), and abab.
2
Evaluate Statement (1) independently.
a−b=4a - b = 4
Knowing a−b=4a - b = 4 leaves a2+ab+b2a^2 + ab + b^2 unknown. For example, if a=4,b=0a=4, b=0, then a3−b3=64a^3-b^3=64. If a=5,b=1a=5, b=1, then a3−b3=124a^3-b^3=124. Multiple values exist, so Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
a2+b2=26a^2 + b^2 = 26
If a=5,b=1a=5, b=1, then a3−b3=124a^3-b^3=124. If a=26,b=0a=\sqrt{26}, b=0, then a3−b3=2626a^3-b^3=26\sqrt{26}. Multiple values exist, so Statement (2) alone is insufficient.
4
Evaluate both statements combined.
ab=5ab = 5 and a3−b3=124a^3 - b^3 = 124
From Statement (1), (a−b)2=a2−2ab+b2=42=16(a - b)^2 = a^2 - 2ab + b^2 = 4^2 = 16. Substituting Statement (2) into this equation gives 26−2ab=16  ⟹  2ab=10  ⟹  ab=526 - 2ab = 16 \implies 2ab = 10 \implies ab = 5. Now substituting all values into the factored identity gives a3−b3=4×(26+5)=4×31=124a^3 - b^3 = 4 \times (26 + 5) = 4 \times 31 = 124. This gives a single, unique value.

Key Concept

Algebraic Factoring of Difference of Cubes and System Solvability
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