Solving Simultaneous Equations: A Simple Example with Numbers Summing to 66

Simultaneous equations are a fundamental concept in algebra and number theory. They involve finding the values of variables that satisfy multiple equations at the same time. In this article, we will solve a specific example where the sum of two numbers is 66, and their difference is 12.

Problem Statement

We are given two equations to solve:

x y 66 x - y 12

Solving the Equations

To solve these simultaneous equations, we can use the method of elimination. First, let's add the two equations together:

(x   y)   (x - y)  66   12
2x  78
x  39

Now that we have the value of x, we can substitute it back into one of the original equations to find y. Using the first equation:

x   y  66
39   y  66
y  66 - 39
y  27

We have successfully solved for x and y. Let's verify our solution:

x y 66 — 39 27 66 (satisfies the first equation) x - y 12 — 39 - 27 12 (satisfies the second equation)

Alternative Methods

There are multiple ways to solve this problem. Here are a couple of alternative methods:

Method 1: Simplifying the Equations

x   y  66
2
33
x - y  12
2
6
2x  78
x  39
y  66 - x  27

Method 2: Substitution

Another method involves expressing one variable in terms of the other:

x y 66 — x 66 - y x - y 12 — (66 - y) - y 12

Simplifying the substitution:

66 - 2y 12 2y 54 y 27 x 66 - 27 39

Once again, we have the same solution for x 39 and y 27.

Verification using Symmetry

To verify, we can also think about the symmetry in the problem:

Average of the numbers: 66 / 2 33 Half the difference: 12 / 2 6 One of the numbers: 33 - 6 27 The other number: 33 6 39

Indeed, this gives us the same solution: x 39 and y 27.

Conclusion

Simultaneous equations can often be solved using various methods, and the solution to the problem presented is x 39 and y 27. These numbers add up to 66 and their difference is 12. This type of problem is a common exercise in algebra and can help in understanding the relationships between numbers and equations.