Of course, let's solve the system of equations step-by-step.
Given system:
[tex]\[
\begin{cases}
3x + 2y - z + 5w = 20 \\
y = 2z - 3w \\
z = w + 1 \\
2w = 8
\end{cases}
\][/tex]
### Step 1: Solve for \( w \)
Start with the last equation:
[tex]\[
2w = 8
\][/tex]
Divide both sides by 2:
[tex]\[
w = 4
\][/tex]
### Step 2: Solve for \( z \)
Next, use the value of \( w \) to find \( z \) from the third equation:
[tex]\[
z = w + 1
\][/tex]
Substitute \( w = 4 \):
[tex]\[
z = 4 + 1 = 5
\][/tex]
### Step 3: Solve for \( y \)
Now, use the values of \( z \) and \( w \) to find \( y \) from the second equation:
[tex]\[
y = 2z - 3w
\][/tex]
Substitute \( z = 5 \) and \( w = 4 \):
[tex]\[
y = 2 \times 5 - 3 \times 4 = 10 - 12 = -2
\][/tex]
### Step 4: Solve for \( x \)
Finally, use the values of \( y \), \( z \), and \( w \) to find \( x \) from the first equation:
[tex]\[
3x + 2y - z + 5w = 20
\][/tex]
Substitute \( y = -2 \), \( z = 5 \), and \( w = 4 \):
[tex]\[
3x + 2(-2) - 5 + 5(4) = 20
\][/tex]
Simplify the equation:
[tex]\[
3x - 4 - 5 + 20 = 20
\][/tex]
Combine the constants:
[tex]\[
3x + 11 = 20
\][/tex]
Subtract 11 from both sides:
[tex]\[
3x = 9
\][/tex]
Divide by 3:
[tex]\[
x = 3
\][/tex]
### Final Solution
The solution to the system is:
[tex]\[
(x, y, z, w) = (3, -2, 5, 4)
\][/tex]
