Find the value of C = x² − 8xy + 16y² given that x − 4y = 5.

Step-by-step solution

1. 1

   Notice that the given condition does not determine x and y separately. One equation in two unknowns has infinitely many solutions, so the expression must be arranged to depend only on the combination $x - 4y$. That is a strong hint to look for a perfect square.

2. 2

   Match the expression to the perfect-square pattern. The identity is

   $(A - B)^2 = A^2 - 2AB + B^2$

   Here $A^2 = x^2$ gives $A = x$, and $B^2 = 16y^2$ gives $B = 4y$. Check the middle term: $2AB = 2 \cdot x \cdot 4y = 8xy$, which matches the $-8xy$ in the expression.

3. 3

   Rewrite the expression as a single square.

   $C = x^2 - 8xy + 16y^2 = (x - 4y)^2$

4. 4

   Substitute the given value of the whole bracket. Since $x - 4y = 5$:

   $C = 5^2 = 25$

   The substitution is of the entire expression $x - 4y$, not of individual values of $x$ and $y$ — that is what makes the problem solvable.

5. 5

   Confirm with a concrete pair. Any $(x, y)$ with $x - 4y = 5$ should give the same answer. Take $y = 0$, $x = 5$: $25 - 0 + 0 = 25 \ \checkmark$. Take $y = 1$, $x = 9$: $81 - 72 + 16 = 25 \ \checkmark$. Take $y = -2$, $x = -3$: $9 - 48 + 64 = 25 \ \checkmark$.