the difference between two numbers is 3 and the difference between their squares is 63. which is the larger number?
What is the larger number if the difference between two numbers is 3 and the difference between their squares is 63?
Answer:
To solve this problem, we need to define and use algebraic expressions that represent the given conditions. Let’s break the problem down into a step-by-step mathematical approach.
Step 1: Define the Variables
Let’s denote the two numbers as x and y, with the condition that x > y.
Step 2: Set Up the Equations Based on the Given Information
According to the problem, we have two conditions:
-
The difference between the two numbers:
$$x - y = 3 \quad \text{(Equation 1)}$$ -
The difference between their squares:
$$x^2 - y^2 = 63 \quad \text{(Equation 2)}$$
Step 3: Use the Difference of Squares Formula
To simplify Equation 2, use the difference of squares formula:
$$x^2 - y^2 = (x + y)(x - y)$$
Plug this into Equation 2:
$$(x + y)(x - y) = 63$$
Step 4: Substitute Known Values
We know from Equation 1 that x - y = 3. Substitute this into the modified Equation 2:
$$(x + y)(3) = 63$$
Solve for x + y:
$$x + y = \frac{63}{3} = 21 \quad \text{(Equation 3)}$$
Step 5: Solve Simultaneous Equations
Now we have a system of two linear equations from Equation 1 and Equation 3:
- x - y = 3
- x + y = 21
To solve this system, add the two equations:
(x - y) + (x + y) = 3 + 21
2x = 24
Now, solve for x:
x = \frac{24}{2} = 12
Now, substitute x = 12 back into Equation 1 to solve for y:
12 - y = 3
y = 12 - 3 = 9
Step 6: Identify the Larger Number
We found x = 12 and y = 9. Since we defined x as the larger number, x = 12 is indeed larger than y = 9.
Final Answer:
The larger number is 12.