Solving the Equation 2x y 6 and x - y 3

When dealing with algebraic equations, it is often necessary to solve systems of simultaneous equations. In this article, we will guide you through the process of solving the following system of equations:

Step-by-Step Solution

The given equations are:

1. 2x y 6

2. x - y 3

Step 1: Equating the Equations

One method to solve this system is to add or subtract the equations to eliminate one of the variables. Let's start by adding the two equations together:

2x y (x - y) 6 3

3x 9

x 3

Step 2: Solving for y

Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the second equation:

x - y 3

3 - y 3

y 0

Step 3: Verification

To ensure that our solution is correct, we can substitute the values of x and y back into the original equations:

2x y 6 becomes 2(3) 0 6, which is true. x - y 3 becomes 3 - 0 3, which is also true.

Therefore, the solution to the system of equations is x 3 and y 0.

Types of Systems of Equations

Systems of equations can be categorized based on their solutions:

Consistent System: The system is consistent when the equations have at least one solution. This is the case with the given system. Inconsistent System: The system is inconsistent when the equations have no solution. An example would be 2x y 6 and x - y 5, as the calculations would lead to a contradiction. Dependent System: The system is dependent when the equations represent the same line. An example is 2x y 6 and 4x 2y 12.

Conclusion

In conclusion, solving the system of equations 2x y 6 and x - y 3 yields the solution x 3 and y 0. By understanding the processes involved, you can apply similar techniques to solve other systems of equations with confidence.