Question

\( (y + 3x - 13)^3 + (x + y - 7)^2 = 0 \) where x and y are integers. The value of \(x^3 + y^3\) is _____.

Hint:

When faced with an equation of the form \(f(x,y)^a + g(x,y)^b = 0\) with integer solutions, especially if one exponent is even, first check the simple case where both \(f(x,y)=0\) and \(g(x,y)=0\). This often leads to the intended unique integer solution.

Answer 1

Step 1: Understanding the Question:
We are given an equation with two variables, \(x\) and \(y\), which are known to be integers. We need to find the value of the expression \(x^3 + y^3\).
Step 2: Key Insight and Approach:
Let \(A = y + 3x - 13\) and \(B = x + y - 7\). The equation becomes \(A^3 + B^2 = 0\).
Since \(x\) and \(y\) are integers, \(A\) and \(B\) must also be integers. The term \(B^2\) is the square of an integer, so it must be non-negative (\(B^2 \ge 0\)).
The equation can be written as \(A^3 = -B^2\). Since \(B^2 \ge 0\), we have \(-B^2 \le 0\). This implies \(A^3 \le 0\), which means \(A \le 0\).
Furthermore, for \(A^3 = -B^2\) to hold for integers, \(|A^3| = B^2\). This means \(|A|\) must be a perfect square and \(|B|\) must be a perfect cube that satisfy the relation.
For example, if \(A=-1\), \(B^2 = -(-1)^3 = 1\), so \(B=\pm 1\). This leads to integer solutions for x and y.
Step 3: Solving the System of Equations:
Assume the unique intended solution comes from setting both terms to zero: \[ y + 3x - 13 = 0 \quad \text{(Equation 1)} \] \[ x + y - 7 = 0 \quad \text{(Equation 2)} \] This is a system of two linear equations. We can solve it by substitution or elimination. Subtracting Equation 2 from Equation 1: \[ (y + 3x - 13) - (x + y - 7) = 0 - 0 \] \[ 2x - 6 = 0 \] \[ 2x = 6 \implies x = 3 \] Now substitute \(x=3\) back into Equation 2: \[ 3 + y - 7 = 0 \] \[ y - 4 = 0 \implies y = 4 \] The solution is \(x=3, y=4\). These are integers, so the solution is valid. Step 4: Calculate the Required Value:
We need to find the value of \(x^3 + y^3\). \[ x^3 + y^3 = (3)^3 + (4)^3 = 27 + 64 = 91 \] Step 5: Final Answer:
The value of \(x^3 + y^3\) is 91.