Solve the equation `3x-5=7-x` by the trial-and-error method.

Answer

To solve the equation \(3x - 5 = 7 - x\) using the trial-and-error method, we will follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ 3x - 5 = 7 - x \]

### Step 2: Create a chart for trial values
We will create a chart to test different values of \(x\) to see when both sides of the equation are equal.

| \(x\) | Left Hand Side (\(3x - 5\)) | Right Hand Side (\(7 - x\)) |
|-------|------------------------------|------------------------------|
| 1 | \(3(1) - 5 = 3 - 5 = -2\) | \(7 - 1 = 6\) |
| 2 | \(3(2) - 5 = 6 - 5 = 1\) | \(7 - 2 = 5\) |
| 3 | \(3(3) - 5 = 9 - 5 = 4\) | \(7 - 3 = 4\) |

### Step 3: Test different values of \(x\)
1. **For \(x = 1\)**:
- Left Hand Side: \(3(1) - 5 = 3 - 5 = -2\)
- Right Hand Side: \(7 - 1 = 6\)
- Not equal.

2. **For \(x = 2\)**:
- Left Hand Side: \(3(2) - 5 = 6 - 5 = 1\)
- Right Hand Side: \(7 - 2 = 5\)
- Not equal.

3. **For \(x = 3\)**:
- Left Hand Side: \(3(3) - 5 = 9 - 5 = 4\)
- Right Hand Side: \(7 - 3 = 4\)
- Equal.

### Step 4: Conclusion
Since both sides are equal when \(x = 3\), we conclude that the solution to the equation \(3x - 5 = 7 - x\) is:
\[ x = 3 \]