`{:("Column A", 7x + 3y = 12","3x + 7y = 8 , "Column B"),(x-y,,1):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the equations \(7x + 3y = 12\) and \(3x + 7y = 8\) and compare \(x - y\) with \(1\), we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(7x + 3y = 12\) (Equation 1)
2. \(3x + 7y = 8\) (Equation 2)

### Step 2: Make the coefficients of \(x\) the same
To eliminate \(x\), we can multiply Equation 1 by \(3\) and Equation 2 by \(7\):

- Multiply Equation 1 by \(3\):
\[ 3(7x + 3y) = 3(12) \implies 21x + 9y = 36 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by \(7\):
\[ 7(3x + 7y) = 7(8) \implies 21x + 49y = 56 \quad \text{(Equation 4)} \]

### Step 3: Subtract the two new equations
Now, we subtract Equation 3 from Equation 4:
\[ (21x + 49y) - (21x + 9y) = 56 - 36 \]
This simplifies to:
\[ 40y = 20 \]

### Step 4: Solve for \(y\)
Now, divide both sides by \(40\):
\[ y = \frac{20}{40} = 0.5 \]

### Step 5: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute \(y = 0.5\) back into one of the original equations. We'll use Equation 2:
\[ 3x + 7(0.5) = 8 \]
This simplifies to:
\[ 3x + 3.5 = 8 \]
Now, subtract \(3.5\) from both sides:
\[ 3x = 8 - 3.5 \implies 3x = 4.5 \]

### Step 6: Solve for \(x\)
Now, divide both sides by \(3\):
\[ x = \frac{4.5}{3} = 1.5 \]

### Step 7: Calculate \(x - y\)
Now we find \(x - y\):
\[ x - y = 1.5 - 0.5 = 1 \]

### Step 8: Compare \(x - y\) with \(1\)
We see that:
\[ x - y = 1 \quad \text{and} \quad \text{Column B} = 1 \]
Thus, \(x - y\) is equal to \(1\).

### Conclusion
Both Column A and Column B are equal, so the final answer is \(C\) (both are equal).

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