If x + y = 12, then the value of (x - 5)^3 + (y - 7)^3 is

Appeared in: UPPSC 2017

Explanation

The problem can be simplified using substitution. Let a = (x - 5) and b = (y - 7).
By adding these new variables, we get a + b = (x - 5) + (y - 7) = x + y - 12.
Using the given information that x + y = 12, the sum a + b becomes 12 - 12 = 0.
The expression to be evaluated is a³ + b³. Since a + b = 0, it means a = -b.
Substituting a = -b into the expression gives (-b)³ + b³ = -b³ + b³ = 0.

Option A: 144 is the square of 12 (12²), which is the value of x + y. This is an incorrect application of the given numbers.
Option B: 125 is the cube of 5 (5³). This might be chosen by mistakenly focusing on the '(x-5)' part of the expression, but it ignores the relationship with '(y-7)' and the initial condition.
Option C: 81 is 9² or 3⁴. There is no mathematical path from the given equations to this result. It is a random distractor.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving algebraic expressions using substitution and identities as background academic context rather than a clinical decision trigger.
This is a general mathematics question testing algebraic skills, which are part of the aptitude section in some nursing entrance exams.
Strong analytical and problem-solving skills are valuable in nursing for tasks like dosage calculations and interpreting data, although this specific algebraic problem does not have a direct clinical application.
What if the equation was x + y = 13? Then a + b = (x+y) - 12 = 13 - 12 = 1. The expression a³ + b³ would no longer be zero and would depend on the specific values of x and y.

How to Approach the Question

First, carefully read the given equation (x + y = 12) and the expression you need to evaluate ((x - 5)³ + (y - 7)³).
Notice the structure of the expression. The terms inside the cubes, (x - 5) and (y - 7), seem related to the initial equation.
Use substitution to simplify the problem. Let a = x - 5 and b = y - 7.
Try to find a relationship between 'a' and 'b'. Calculate their sum: a + b = (x - 5) + (y - 7) = x + y - 12.
Substitute the value from the given equation (x + y = 12) into the sum you just found: a + b = 12 - 12 = 0.
Now, evaluate the simplified expression a³ + b³ using the fact that a + b = 0 (which implies a = -b). The result will be 0.

Concept Tested & Keywords

Concept Tested: Solving algebraic expressions using substitution and identities.
Stem keywords: x + y = 12, algebraic expression, sum of cubes
Lead-in keywords: value of
Negative lead-in flag: false

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