Question

Find the values of x, y, and λ that satisfy the system of equations. Such systems arise in certain problems of calculus, and λ is called the Lagrange multiplier. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set λ = t and solve for x and y in terms of t.) 4x + λ = 0 4y + λ = 0 x + y − 8 = 0

Answer 1

Answer:

x=4, y=4, λ=-16

Step-by-step explanation:

We have this 3x3 system of linear equations:

$4x+$λ$=0$

$4y+$λ$=0$

$x+y=8$

So, let's rewrite the system in its augmented matrix form

$\left[\begin{array}{cccc}4&0&1&0\\0&4&1&0\\1&1&0&8\end{array}\right]$

Let´s apply row reduction process to  its associated augmented matrix:

Swap R1 and R3

$\left[\begin{array}{cccc}1&1&0&8\\0&4&1&0\\4&0&1&0\end{array}\right]$

R3-4R1

$\left[\begin{array}{cccc}1&1&0&8\\0&4&1&0\\0&-4&1&-32\end{array}\right]$

R3+R2

$\left[\begin{array}{cccc}1&1&0&8\\0&4&1&0\\0&0&2&-32\end{array}\right]$

Now we have a simplified system:

x+y+0=0

0+4y+λ=0

0+0+2λ=-32

Solving for λ, x, and y

λ=-16

x=4

y=4