Answer :
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]
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]
