Find the smallest value of the expression (x + 2y) ^ 2 + (x + y-1) ^ 2 and the x and y values at which it is reached.

The expression (x + 2 * y) ^ 2 + (x + y – 1) ^ 2 is given. The expression is positive, since (x + 2 * y) ^ 2 and (x + y – 1) ^ 2 are not negative, but positive. Then, the expression (x + 2 * y) ^ 2 + (x + y – 1) ^ 2 has the smallest value if the expression (x + 2 * y) ^ 2 = 0 and (x + y – 1) ^ 2 = 0.
We get the system:
x + 2 * y = 0;
x + y – 1 = 0;
1) From the first equation x = – 2 * y.
2) Substitute in the second equation:
– 2 * y + y – 1 = 0;
– y -1 = 0;
-y = 1;
y = – 1;
3) x = – 2 * y = – 2 * (- 1) = 2;
Answer: x = 2, y = – 1.