Question

Find z, if it is known that: a: $-y^2 + x^2 = 20$ b: $y^3 - 2x^2 - 4z \ge -12$ and c: $x, y$ and $z$ are all positive integers

Accepted Answer

E. From (a): $(x-y)(x+y) = 20$. Since $x, y$ positive integers, factors of 20 with same parity are $(x+y=10, x-y=2) \implies x = 6, y = 4$. Substitute into (b): $4^3 - 2(6^2) - 4z \ge -12 \implies 64 - 72 - 4z \ge -12 \implies -8 - 4z \ge -12 \implies 4z \le 4 \implies z \le 1$. Since $z$ is a positive integer, $z = 1$.

Positive Integer System of Equations

Explanation

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