Solving Linear Equations through System of Equations: A Practical Example

Linear equations form a core part of algebra and are widely used to model real-world situations. In this article, we will solve a problem involving the sum and difference of two numbers using a system of equations. This will provide a practical example of how to approach and solve such problems.

Problem Statement

In this problem, we are given two conditions for two numbers:

The sum of the two numbers is 7. If one number is subtracted from the other, their difference is -11.

Let's denote these two numbers by x and y. We can then translate the given conditions into a system of linear equations:

x y 7 x - y -11

Solving the System of Equations

Solve for one variable: From the first equation, we can express y in terms of x:

y 7 - x

Substitute into the second equation: Substitute y into the second equation:

x - (7 - x) -11

x - 7 x -11

2x - 7 -11

Solve for x: Add 7 to both sides:

2x -4

x -2

Solve for y: Now that we have x -2, we can find y:

y 7 - (-2) 7 2 9

Conclusion

The two numbers are x -2 and y 9. We can now verify the solution:

x y -2 9 7 x - y -2 - 9 -11

Thus, the numbers are -2 and 9.

Additional Information and Checking the Solution

We can also solve the problem using another method:

Determine the average of the sum and the difference:

7/2 3.5 and -11/2 -5.5

One number is 3.5 - (-5.5) 9, and the other is 3.5 - 5.5 -2

Use the second method to verify the solution:

ab 7 and a - b -11

Solve for a:

2a -4 or a -2

b -2 11 9

Therefore, the numbers are -2 and 9.