If `a-b=3 and a^(2)-b^(2)=21`, what is the value of a?

Options

A. `-3`
B. `-2`
C. `2`
D. `5`

Answer

To solve the equations \( a - b = 3 \) and \( a^2 - b^2 = 21 \), we can follow these steps:

### Step 1: Use the difference of squares
We know that \( a^2 - b^2 \) can be factored as:
\[ a^2 - b^2 = (a - b)(a + b) \]
Given that \( a - b = 3 \), we can substitute this into the equation:
\[ (a - b)(a + b) = 21 \]
This becomes:
\[ 3(a + b) = 21 \]

### Step 2: Solve for \( a + b \)
Now, we can solve for \( a + b \) by dividing both sides by 3:
\[ a + b = \frac{21}{3} = 7 \]

### Step 3: Set up a system of equations
Now we have a system of two equations:
1. \( a - b = 3 \)
2. \( a + b = 7 \)

### Step 4: Add the two equations
To eliminate \( b \), we can add the two equations:
\[ (a - b) + (a + b) = 3 + 7 \]
This simplifies to:
\[ 2a = 10 \]

### Step 5: Solve for \( a \)
Now, divide both sides by 2 to find \( a \):
\[ a = \frac{10}{2} = 5 \]

### Conclusion
Thus, the value of \( a \) is:
\[ \boxed{5} \]