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

x=4, y=4, λ=-16

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