Solving and Analyzing the Inconsistency of Equations x - y 20 and x - y -20

The equations x - y 20 and x - y -20 present a classic example of inconsistency in a system of linear equations. Let us delve into the details of why there are no solutions for x and y that satisfy both equations simultaneously.

Understanding Inconsistent Equations

Two equations are deemed to be inconsistent if they cannot be satisfied together. In this case, the equations x - y 20 and x - y -20 directly contradict each other, as any constant term cannot simultaneously be 20 and -20.

Step-by-Step Analysis

First, let's solve the equations step by step to understand why they are inconsistent:

Method 1: Direct Comparison

The first equation is:

x - y  20             (1)

And the second equation is:

x - y  -20             (2)

Since both equations state that x - y equals two different values, it is geometrically clear that these lines are parallel and never intersect, hence no solutions.

Another way to look at it mathematically is to add and subtract the two equations:

x - y   x - y  20 - 202x - 2y   - y  0

This simplifies to x y, suggesting that for the equations to be consistent, x should equal y. However, substituting y for x in x - y 20 and x - y -20 results in:

y - y  20 and y - y  -200  20 and 0  -20

Clearly, this is a contradiction, indicating that the given equations are inconsistent.

Method 2: Graphical Analysis

Another method to verify the inconsistency is through graphing both equations on the same coordinate plane:

The equations x - y 20 and x - y -20 can be rewritten in the form y x - 20 and y x 20. These equations represent two parallel lines that never intersect. The slope of both lines is 1, and they have y-intercepts at -20 and 20, respectively.

Conclusion: The lines are parallel, and therefore there is no pair of (x, y) that satisfies both equations simultaneously.

Conclusion

From the analysis, it is evident that the system of equations x - y 20 and x - y -20 is inconsistent. This means that there are no values of x and y that can satisfy both equations at the same time. In mathematical terms, the system of equations is unsolvable.