Questions: Solve for (x, y, z) and (w). (Enter your answers as a comma-separated list.) [ left[beginarraycc x+y y+z z+w w endarrayright]=left[beginarrayll 3 5 5 2 endarrayright] ] ((x, y, z, w)=)

Solve for (x, y, z) and (w). (Enter your answers as a comma-separated list.)

[
left[beginarraycc
x+y y+z
z+w w
endarrayright]=left[beginarrayll
3 5
5 2
endarrayright]
]

((x, y, z, w)=)

Transcript text: Solve for $x, y, z$ and $w$. (Enter your answers as a comma-separated list.) \[ \begin{array}{r} {\left[\begin{array}{cc} x+y & y+z \\ z+w & w \end{array}\right]=\left[\begin{array}{ll} 3 & 5 \\ 5 & 2 \end{array}\right]} \\ (x, y, z, w)=(\square \end{array} \]

Solution

Solution Steps

Solution Approach

To solve for $x, y, z,$ and $w$, we need to equate the corresponding elements of the two matrices. This will give us a system of equations:

$x + y = 3$
$y + z = 5$
$z + w = 5$
$w = 2$

We can solve this system of equations step-by-step to find the values of $x, y, z,$ and $w$.

Step 1: Set Up the Equations

We start with the matrix equation given in the problem, which leads to the following system of equations:

$ x + y = 3 $
$ y + z = 5 $
$ z + w = 5 $
$ w = 2 $

Step 2: Solve for $ w $

From equation 4, we directly find: \[ w = 2 \]

Step 3: Substitute $ w $ into the Third Equation

Substituting $ w = 2 $ into equation 3: \[ z + 2 = 5 \] This simplifies to: \[ z = 3 \]

Step 4: Substitute $ z $ into the Second Equation

Now, substituting $ z = 3 $ into equation 2: \[ y + 3 = 5 \] This simplifies to: \[ y = 2 \]

Step 5: Substitute $ y $ into the First Equation

Finally, substituting $ y = 2 $ into equation 1: \[ x + 2 = 3 \] This simplifies to: \[ x = 1 \]

Final Answer

The values of $ x, y, z, $ and $ w $ are: \[ \boxed{(x, y, z, w) = (1, 2, 3, 2)} \]

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