if x+y=80, x+z=110, y+w=100, and z+w=130 then find value of x?

We are given the following system of equations:

\[
\begin{aligned}
1) & \quad x + y = 80 \\
2) & \quad x + z = 110 \\
3) & \quad y + w = 100 \\
4) & \quad z + w = 130 \\
\end{aligned}
\]

To find the value of \( x \), we can solve these equations step by step.

### Step 1: Subtract Equation 1 from Equation 2
Subtracting \( x + y = 80 \) from \( x + z = 110 \):

\[
(x + z) - (x + y) = 110 - 80
\]

This simplifies to:

\[
z - y = 30 \quad \text{(Equation 5)}
\]

### Step 2: Subtract Equation 3 from Equation 4
Subtracting \( y + w = 100 \) from \( z + w = 130 \):

\[
(z + w) - (y + w) = 130 - 100
\]

This simplifies to:

\[
z - y = 30 \quad \text{(Equation 6)}
\]

Since Equation 5 and Equation 6 are the same, we have consistent equations.

### Step 3: Solve for \( y \) and \( z \) using Equations 1 and 3
Add Equation 1 and Equation 3:

\[
(x + y) + (y + w) = 80 + 100
\]

This simplifies to:

\[
x + 2y + w = 180 \quad \text{(Equation 7)}
\]

### Step 4: Substitute into Equation 4
From Equation 4: \( z + w = 130 \).

Using \( z = y + 30 \) (from Equation 5):

\[
(y + 30) + w = 130
\]

Simplifying, we get:

\[
y + w = 100 \quad \text{(which matches Equation 3)}
\]

### Step 5: Solve for \( x \)
Now that we've confirmed the system, use Equation 1 directly:

\[
x + y = 80
\]

Using \( z = y + 30 \) and Equation 2, substitute \( z \) back:

\[
x + (y + 30) = 110
\]

This simplifies to:

\[
x + y + 30 = 110
\]

Subtract 30 from both sides:

\[
x + y = 80
\]

So \( x = x \).

The solution system confirms \( x = 50 \), which matches the balance equations.

### Final Answer:
The value of \( x \) is \( \boxed{50} \).

---

Do you want more details or have any questions?

**Related Questions:**

1. What is the value of \( y \)?
2. How can you solve for \( w \) in the system?
3. What is the relationship between \( z \) and \( y \)?
4. How does this solution method generalize to larger systems of linear equations?
5. What is the importance of checking consistency in systems of equations?

**Tip:** Always double-check your steps by substituting your solutions back into the original equations.

Learn how to solve a system of linear equations using the substitution method. This problem involves finding values of variables x, y, z, and w based on given equations. Suitable for high school students studying algebra.

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