Solving Two-Number Problems Using Simultaneous Equations

In this article, we will explore a common type of algebra problem where the difference and the sum of two numbers are given. We will learn how to solve such problems using simultaneous equations and algebraic manipulation. This method is useful for various applications, including real-world scenarios and competitive mathematics.

Introduction to the Problem

The problem statement typically provides two conditions: the difference between two numbers and their sum. For example, the difference between two numbers is 3 and their sum is 13. Our task is to find the two numbers that satisfy both conditions.

Step-by-Step Solution

Let's denote the two numbers as x and y.

Step 1: Define the equations based on the given conditions:

x - y 3 (Equation 1)

x y 13 (Equation 2)

Step 2: Solve one of the equations for one variable. Let's solve Equation 1 for y:

y x - 3

Step 3: Substitute the expression for y from Step 2 into Equation 2:

x (x - 3) 13

Step 4: Simplify and solve for x:

2x - 3 13

2x 16

x 8

Step 5: Substitute x 8 back into the expression for y:

y 8 - 3

y 5

Step 6: Verify the solution:

Check: x y 8 5 13

Verification: x - y 8 - 5 3

Therefore, the two numbers are 8 and 5.

Alternative Method: Using the Average and Difference

Another approach is to use the average of the two numbers and their difference. The average of the two numbers can be found by dividing their sum by 2:

6.5 (x y) / 2 13 / 2

The smaller number is obtained by subtracting half the difference from the average:

Smaller number 6.5 - 1.5 5

The larger number is obtained by adding half the difference to the average:

Larger number 6.5 1.5 8

Solving the Problem Using Simple Algebraic Manipulation

Another method involves adding the two equations together:

2x 16

x 8

Then, substituting x 8 into one of the original equations:

8 y 13

y 5

This confirms the solution as 8 and 5.

Conclusion

Through this detailed exploration, we have demonstrated how to solve a two-number problem given the conditions of their sum and difference using several algebraic methods. The key takeaway is that by forming and solving simultaneous equations, we can accurately find the values of the unknown numbers in such problems.