Let's solve the system of four equations given in your image. The system is:

1. $3x + 2y - z + 5w = 20$
2. $y = 2z - 3w$
3. $z = w + 1$
4. $2w = 8$

Step-by-Step Solution:

Step 1: Solve for $w$ from the fourth equation

$2w = 8 \implies w = 4$

Step 2: Substitute $w = 4$ into the third equation

$z = w + 1 \implies z = 4 + 1 = 5$

Step 3: Substitute $z = 5$ and $w = 4$ into the second equation

$y = 2z - 3w \implies y = 2(5) - 3(4) = 10 - 12 = -2$

Step 4: Substitute $y = -2$, $z = 5$, and $w = 4$ into the first equation

$3x + 2y - z + 5w = 20$ $3x + 2(-2) - 5 + 5(4) = 20$ $3x - 4 - 5 + 20 = 20$ $3x + 11 = 20$ $3x = 9 \implies x = 3$

Final Solution:

$x = 3, \, y = -2, \, z = 5, \, w = 4$

Would you like a detailed explanation of any step, or help solving another system?

Here are 5 related questions to consider:

1. How do we verify the solution for all four equations?
2. What happens if one of the equations is inconsistent with the others?
3. Can this system be solved graphically? Why or why not?
4. How does substitution compare to other methods like elimination or matrices?
5. How can systems of equations like this be applied to real-world problems?

Tip: Always start solving with the simplest