Find values of x,y,and ? that satisfy the system.These systems arise in certain optimization problems in calculus,and ? is called a Lagrange multiplier.? $\left\{\begin{array}{ll}
2 x+\lambda & =0 \
2 y+\lambda & =0 \
x+y-10 & =0
\end{array}\right.$ ?
A)x = 5?,y = -10,? = -10
B)?x = 5,y = 5,? = 5
C)x = -10,y = 5,? = -10
D)x = 5,y = -10,? = 5
E)x = 5,y = 5,? = -10

Question

Quizplus · Accepted Answer

From the first two equations, $2x + \lambda = 0$ and $2y + \lambda = 0$, we can see that $2x = 2y$, which implies $x = y$. From the third equation, $x + y - 10 = 0$, substituting $x = y$, we get $2x = 10$, so $x = 5$. Therefore, $y = 5$ as well. Substituting $x = 5$ into the first equation, $2(5) + \lambda = 0$, we get $\lambda = -10$.

Find Values of X,y,and ? That Satisfy the System {2x+λ=02y+λ=0x+y−10=0\left\{\begin{array}{ll}2 x+\lambda & =0 \\2 y+\lambda & =0 \\x+y-10 & =0\end{array}\right.⎩⎨⎧​2x+λ2y+λx+y−10​=0=0=0​

Find Values of X,y,and ? That Satisfy the System $\left\{\begin{array}{ll}2 x+\lambda & =0 \\2 y+\lambda & =0 \\x+y-10 & =0\end{array}\right.$