https://file.aiquickdraw.com/mathbot/file/1729861717031ut73a8xd.jpg

The system of simultaneous equations provided in the image is as follows:

1. \( x + 2y = 143 \) (Equation 1)
2. \( y + 3W = -74 \) (Equation 2)
3. \( 4x + 5W = 4 \) (Equation 3)

We are tasked with solving for \( x \), \( y \), and \( W \), where \( W \in \mathbb{Z} \) (i.e., \( W \) is an integer).

### Step-by-step solution:

**Step 1: Express \( y \) in terms of \( W \) from Equation (2):**
\[
y = -74 - 3W
\]
Substitute this expression for \( y \) in Equation (1).

**Step 2: Substitute into Equation (1):**
\[
x + 2(-74 - 3W) = 143
\]
Simplify the equation:
\[
x - 148 - 6W = 143
\]
\[
x = 143 + 148 + 6W
\]
\[
x = 291 + 6W
\]
Now we have \( x = 291 + 6W \).

**Step 3: Substitute \( x \) into Equation (3):**
Substitute \( x = 291 + 6W \) into Equation (3):
\[
4(291 + 6W) + 5W = 4
\]
Simplify the equation:
\[
1164 + 24W + 5W = 4
\]
\[
1164 + 29W = 4
\]
Solve for \( W \):
\[
29W = 4 - 1164
\]
\[
29W = -1160
\]
\[
W = \frac{-1160}{29} = -40
\]

**Step 4: Find \( y \) and \( x \):**
Now that we know \( W = -40 \), substitute it back into the expressions for \( y \) and \( x \):

- \( y = -74 - 3(-40) = -74 + 120 = 46 \)
- \( x = 291 + 6(-40) = 291 - 240 = 51 \)

### Final Solution:
The solution to the system is:
\[
x = 51, \quad y = 46, \quad W = -40
\]

Would you like any further clarification or additional details? 

Here are five related questions to deepen your understanding:
1. How can the substitution method be used for more complex systems of equations?
2. How would the solution change if one equation were non-linear?
3. What happens if no integer solution exists for \( W \)?
4. Can you use the matrix method to solve this system of equations?
5. How would you verify the solution is correct?

**Tip**: Always check your solutions by substituting the values back into the original equations to ensure they satisfy all conditions.

Solve the following simultaneous equation for x, y, and W. (W ∈ Z)

Learn how to solve the simultaneous equations for x, y, and W (where W is an integer) using algebraic techniques. This step-by-step guide covers substitution methods and key algebraic concepts suitable for Grades 9-12.

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