The difference of two numbers is 3 and the difference of their square is 69. Find the numbers.

Answer

Let's solve the problem step by step.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
According to the problem:
1. The difference of the two numbers is 3:
\[ y - x = 3 \quad \text{(Equation 1)} \]
2. The difference of their squares is 69:
\[ y^2 - x^2 = 69 \quad \text{(Equation 2)} \]

### Step 3: Factor the Difference of Squares
We can factor the second equation using the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ y^2 - x^2 = (y - x)(y + x) \]
Substituting Equation 1 into this, we have:
\[ (y - x)(y + x) = 69 \]
Since \( y - x = 3 \), we can substitute this into the equation:
\[ 3(y + x) = 69 \]

### Step 4: Solve for \( y + x \)
Now, divide both sides by 3:
\[ y + x = \frac{69}{3} = 23 \quad \text{(Equation 3)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( y - x = 3 \) (Equation 1)
2. \( y + x = 23 \) (Equation 3)

We can add these two equations:
\[ (y - x) + (y + x) = 3 + 23 \]
This simplifies to:
\[ 2y = 26 \]
Now, divide by 2:
\[ y = \frac{26}{2} = 13 \quad \text{(Equation 4)} \]

### Step 6: Substitute to Find \( x \)
Now substitute \( y = 13 \) back into Equation 1 to find \( x \):
\[ 13 - x = 3 \]
Rearranging gives:
\[ x = 13 - 3 = 10 \]

### Step 7: Conclusion
The two numbers are:
\[ x = 10 \quad \text{and} \quad y = 13 \]

### Final Answer
The two numbers are 10 and 13.

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