Solving the Equation: Finding the Two Numbers Based on Sum and Difference
In mathematics, solving systems of equations is a fundamental skill often encountered in algebra. Consider the following scenario: Given that the difference of two numbers is 18 and their sum is 34, we aim to find the values of these numbers. This problem can be approached using basic algebraic principles, which not only enhance analytical skills but also provide a clear understanding of how to manipulate equations.
Problem Statement
We can denote the two numbers as x and y.
Step 1: Setting Up Equations
Based on the problem statement, we establish the following system of equations:
x - y 18
x y 34
Step 2: Solving the Equations
To solve for x and y, we can start by adding the two equations together. This approach simplifies the system to a single equation, making it easier to find one of the variables.
x - y x y 18 34
2x 52
x 26
Having determined that x 26, we then substitute this value back into one of the original equations to solve for y.
26 y 34
y 8
Thus, the two numbers are 26 and 8.
Verification
To confirm our solution, we can verify that both conditions are satisfied:
The difference between 26 and 8 is 18:
26 - 8 18
The sum of 26 and 8 is 34:
26 8 34
Algebraic Manipulation
Alternatively, we can use algebraic manipulation to solve the system of equations. For instance:
x 18 y
Substitute x in the sum equation:
(18 y) y 34
18 2y 34
2y 16
y 8
Substitute y back into the equation for x:
x 18 8 26
Conclusion
This problem demonstrates the importance of basic algebraic principles in solving complex equations. By understanding how to manipulate and combine equations, we can efficiently solve for unknown variables. This skill is not only useful in mathematics but also in various fields such as physics, engineering, and economics, where systems of equations are common tools for problem-solving.
Final Answer
The two numbers are 26 and 8.