Question

Advanced Applications In Exercises 73 and $74 ,$ find values of $x , y ,$ and $\lambda$ that satisfy the system. These systems arise in certain optimization problems in calculus, and $\lambda$ is called a Lagrange multiplier. $$ \left\{ \begin{aligned} 2 + 2 y + 2 \lambda & = 0 \\ 2 x + 1 + \lambda & = 0 \\ 2 x + y - 100 & = 0 \end{aligned} \right. $$

          Advanced Applications In Exercises 73 and $74 ,$ find
values of $x , y ,$ and $\lambda$ that satisfy the system. These systems
arise in certain optimization problems in calculus, and $\lambda$
is called a Lagrange multiplier.

$$
\left\{ \begin{aligned} 2 + 2 y + 2 \lambda & = 0 \\ 2 x + 1 + \lambda & = 0 \\ 2 x + y - 100 & = 0 \end{aligned} \right.
$$

Added by Donna H.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Suzanne Harwood

Step 1

$$ Substituting this into the first equation, we get: $$2 + 2y - 4x - 2 = 0,$$ which simplifies to: $$y = 2x - 1.$$ Substituting this into the third equation, we get: $$2x + (2x - 1) - 100 = 0,$$ which simplifies to: $$x = \frac{101}{4}.$$ Show more…

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Advanced Applications In Exercises 73 and $74 ,$ find values of $x , y ,$ and $\lambda$ that satisfy the system. These systems arise in certain optimization problems in calculus, and $\lambda$ is called a Lagrange multiplier. $$ \left\{ \begin{aligned} 2 + 2 y + 2 \lambda & = 0 \\ 2 x + 1 + \lambda & = 0 \\ 2 x + y - 100 & = 0 \end{aligned} \right. $$