If `999x + 888y = 1332 and 888x + 999y = 555, ` then `x ^(2) - y ^(2) ` is equal to:

Options

A. 7
B. 8
C. 9
D. 5

Answer

To solve the equations \(999x + 888y = 1332\) and \(888x + 999y = 555\), we will follow these steps:

### Step 1: Write down the equations
We have:
1. \(999x + 888y = 1332\) (Equation 1)
2. \(888x + 999y = 555\) (Equation 2)

### Step 2: Add both equations
Adding Equation 1 and Equation 2:
\[ (999x + 888y) + (888x + 999y) = 1332 + 555 \]
This simplifies to:
\[ 1887x + 1887y = 1887 \]

### Step 3: Factor out the common term
We can factor out \(1887\) from the left side:
\[ 1887(x + y) = 1887 \]

### Step 4: Divide both sides by 1887
Dividing both sides by \(1887\):
\[ x + y = 1 \quad \text{(Equation 3)} \]

### Step 5: Subtract Equation 2 from Equation 1
Now, we will subtract Equation 2 from Equation 1:
\[ (999x + 888y) - (888x + 999y) = 1332 - 555 \]
This simplifies to:
\[ (999x - 888x) + (888y - 999y) = 777 \]
Which further simplifies to:
\[ 111x - 111y = 777 \]

### Step 6: Factor out the common term
Factoring out \(111\):
\[ 111(x - y) = 777 \]

### Step 7: Divide both sides by 111
Dividing both sides by \(111\):
\[ x - y = 7 \quad \text{(Equation 4)} \]

### Step 8: Use the results to find \(x^2 - y^2\)
We know that:
\[ x^2 - y^2 = (x - y)(x + y) \]
Substituting the values from Equations 3 and 4:
\[ x^2 - y^2 = (7)(1) = 7 \]

### Final Answer
Thus, \(x^2 - y^2 = 7\).

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