Solving the Problem: The Difference of Two Numbers is -46 and Their Sum is 36

Today, we will explore a classic algebraic problem involving the sum and difference of two numbers. Specifically, we know that the difference of two numbers is -46, and their sum is 36. Our goal is to determine what these two numbers are. We'll solve this problem step-by-step and present two different methods to approach this question.

Method 1: Using Simultaneous Equations

Let the two numbers be x and y. From the problem, we have the following two equations:

Equation 1: The Difference of the Numbers

x - y -46

Equation 2: The Sum of the Numbers

x y 36

We can solve these equations step by step.

Step 1: Solve for One Variable

From the first equation, we can express x in terms of y: x y - 46

Step 2: Substitute into the Second Equation

Now substitute x in the second equation: (y - 46) y 36

Step 3: Simplify and Solve for y

Combine like terms: 2y - 46 36

Add 46 to both sides:

2y 82

Now divide by 2:

y 41

Step 4: Substitute Back to Find x

Now that we have y, substitute it back into the expression for x: x y - 46 41 - 46 -5

Conclusion

The two numbers are:

x -5

y 41

Thus, the two numbers are -5 and 41.

Method 2: Heuristics Approach

In this method, we don't use equations but rather split the difference and the sum. We start by finding half of the sum and half of the difference. Once we have these, we can easily determine the two numbers.

Half the Sum and Half the Difference

Half the sum is 18 and half the difference is 23. The sign of the difference is irrelevant.

Half the sum plus half the difference is 41:

(18 23) 41

Half the sum minus half the difference is -5:

(18 - 23) -5

Essentially, if you reverse the sign of the difference, you still get the same two numbers -5 and 41.

The two numbers are -5 and 41.

To verify our solution, let's check the conditions:

The sum of -5 and 41 is 36:

-5 41 36

The difference between 41 and -5 is -46:

41 - (-5) 41 5 46 (we take the negative difference)

Both conditions are satisfied, confirming that our solution is correct.

This problem teaches us the importance of algebraic methods and heuristics in solving real-world problems. Understanding these methods can help us tackle similar problems more efficiently and effectively.