If the difference of two numbers is 3 and the difference of their squares is 39, then the larger number is

Options

A. 8
B. 9
C. 12
D. 13

Answer

To solve the problem step by step, we will use the information given in the question.

### Step 1: Define the Variables
Let the larger number be \( x \) and the smaller number be \( y \).

### Step 2: Set Up the Equations
From the problem, we have two pieces of information:
1. The difference of the two numbers is 3:
\[ x - y = 3 \quad \text{(Equation 1)} \]
2. The difference of their squares is 39:
\[ x^2 - y^2 = 39 \quad \text{(Equation 2)} \]

### Step 3: Use the Difference of Squares Identity
We can use the identity for the difference of squares:
\[ x^2 - y^2 = (x - y)(x + y) \]
Substituting Equation 1 into this identity:
\[ x^2 - y^2 = (3)(x + y) = 39 \]

### Step 4: Solve for \( x + y \)
From the equation \( 3(x + y) = 39 \), we can solve for \( x + y \):
\[ x + y = \frac{39}{3} = 13 \quad \text{(Equation 3)} \]

### Step 5: Solve the System of Equations
Now we have a system of two equations:
1. \( x - y = 3 \) (Equation 1)
2. \( x + y = 13 \) (Equation 3)

We can add these two equations together:
\[ (x - y) + (x + y) = 3 + 13 \]
This simplifies to:
\[ 2x = 16 \]
So, we can solve for \( x \):
\[ x = \frac{16}{2} = 8 \]

### Step 6: Find the Larger Number
Since we defined \( x \) as the larger number, we conclude that the larger number is:
\[ \text{Larger number} = 8 \]

### Final Answer
The larger number is \( 8 \).

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