Solving Equations of Sum and Difference of Numbers

Given two numbers, let's explore the logic and method of finding these numbers based on their sum and difference, using algebraic equations.

Example Problem:

The sum of two numbers is 38 and their difference is 10. How can we find these numbers?

Step-by-Step Solution:

Let the two numbers be x and y. We can set up the following equations based on the information provided:

x y 38 This equation represents the sum of the two numbers. x - y 10 This equation represents the difference between the two numbers.

To solve for x and y, we can add the two equations together to eliminate y:

x y x - y 38 10

This simplifies to:

2x 48

Dividing both sides by 2 gives:

x 24

Now we can substitute x 24 back into one of the original equations to find y. Using the first equation:

24 y 38

Subtracting 24 from both sides gives:

y 14

Therefore, the two numbers are 24 and 14.

Verification:

Let's verify the solution:

24 14 38 (check sum) 24 - 14 10 (check difference)

Alternative Solutions:

Let's explore a couple of alternative solutions to the same problem to reinforce the concept:

Alternative Solution 1:

Let the two numbers be x and y.

xy 48 …Equation no.1 x - y 12 …Equation no.2

Add the two equations together:

xy (x - y) 48 12

This simplifies to:

2x 60

Dividing both sides by 2 gives:

x 30

Substitute x 30 into any of the original equations. Using Equation no.1:

30y 48

Dividing both sides by 30 gives:

y 1.6

Therefore, the numbers are 30 and 1.6.

Verification:

30 * 1.6 48 (check product) 30 - 1.6 28.4 (check difference)

Alternative Solution 2:

Let the two numbers be x and y.

Given:

xy 38 …Equation no.1 x - y 10 …Equation no.2

Let's add the equations:

xy (x - y) 38 10

This simplifies to:

2x 48

Dividing both sides by 2 gives:

x 24

Substitute x 24 into any of the original equations. Using Equation no.1:

24y 38

Dividing both sides by 24 gives:

y 14

Therefore, the numbers are 24 and 14.

Verification:

24 * 14 336 (Error: Check product) 24 - 14 10 (check difference)

The solution is correct for the difference but incorrect for the product. It’s essential to verify all conditions in such problems.

Conclusion:

Understanding the process of solving equations for the sum and difference of numbers is crucial in algebra. By following a systematic approach, we can accurately determine the numbers involved. This method can be applied in various real-world problems, such as those involving financial calculations, measurements, and more.

Remember, always verify your answers to ensure they meet all given conditions. This helps in avoiding common errors and enhances the reliability of your solutions.