If 4 (x – y) = 64 and 4 (x + y) = 1024, then find the value of x

Concept: When the same positive base (other than 1) is raised to two exponents that give equal values, the exponents themselves must be equal — if am = an then m = n. Rewriting both sides of an exponential equation using the same base turns it into a linear equation in the exponent, and two such linear equations in two unknowns can be solved together by elimination.

Application:

1. Rewrite 64 as a power of 4: 64 = 43, so 4(x – y) = 43 gives the linear equation x − y = 3.

2. Rewrite 1024 as a power of 4: 1024 = 45, so 4(x + y) = 45 gives the linear equation x + y = 5.

3. Add the two linear equations to eliminate y: (x − y) + (x + y) = 3 + 5, which simplifies to 2x = 8, so x = 4.

4. Subtract the two linear equations to find y as well: (x + y) − (x − y) = 5 − 3, which simplifies to 2y = 2, so y = 1.

Cross-check: Substituting x = 4 and y = 1 back into the original expressions confirms both hold: 4(4 – 1) = 43 = 64 and 4(4 + 1) = 45 = 1024, so x = 4 is confirmed.