Question

Розв’язати систему рівнянь x + y = 7 x² + y² = 25. У відповідь записати найбільшу суму x0 + y0, де (x0; y0) – розв’язок системи.

Answer 1

х1у1(3;4)

х2у2(4;3)

........

Answer 2

To solve this system of equations, we can use substitution or elimination. Let's go with substitution. From the first equation, we can express $x$ in terms of $y$:

$x + y = 7 \implies x = 7 - y$

Now substitute this expression for $x$ into the second equation:

$(7 - y)^2 + y^2 = 25$

Expand and simplify:

$49 - 14y + y^2 + y^2 = 25$

Combine like terms:

$2y^2 - 14y + 24 = 0$

Now, we can solve this quadratic equation for $y$. Once we find $y$, we can substitute it back into the expression for $x$ to get the corresponding $x$ value.

Alternatively, we can use the fact that the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-b/a$ and the product of the roots is given by $c/a$.

In our case, the sum of the roots is $-(-14)/2 = 7$ and the product is $24/2 = 12$. Let $y_1$ and $y_2$ be the roots. The sum $x_0 + y_0$ is maximized when $x_0$ and $y_0$ are the roots of the system.

Now, $x_0 + y_0 = 7$. Therefore, the maximum sum is 7.

I hope that helps!