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Understanding the concept of an ordered pair solution is crucial when solving systems of equations. An ordered pair, typically written as (x, y), represents potential values for the variables x and y that may satisfy both equations simultaneously. To verify whether an ordered pair is a solution, we substitute the x and y values into each equation. If both equations are satisfied—meaning they produce true statements—the ordered pair is indeed a solution.

From the exercise given, to determine if (2,1) is a solution, we replaced x with 2 and y with 1 in the equations. However, this did not satisfy the first equation, hence (2,1) is not a solution. Conversely, the ordered pair (2,2) satisfied both equations, thus it is a legitimate solution. Remember that for a pair to qualify as a solution, both equations must hold true after substitution. If either equation does not match, the pair isn't a solution.

The substitution method is a widely used technique for solving a system of equations algebraically. In this method, we express one variable in terms of the other using one of the equations, and then substitute this expression into the other equation. This substitution allows us to solve for one variable. Once we've found the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.

While the original exercise does not require using the substitution method to find the ordered pair solution, it's a good skill to master. To start, you could isolate y from the equation 2x - y = 2 to get y = 2x - 2 and then substitute this into the second equation, x + 3y = 8. This way, you can find the values for x and y that satisfy both equations without guessing or checking potential solutions.

When we talk about an algebraic solution to a system of equations, we refer to solving the system by manipulation of the equations using algebraic methods, such as substitution, elimination, or graphing. The objective is to find the values of the variables that satisfy all equations in the system. These values make up the solution set.

For the system in question, the algebraic solution involves checking given ordered pairs instead of solving for them. However, if we were to solve the system, we would use algebraic methods such as the substitution method mentioned earlier. By applying these techniques, we would arrive at a precise solution for x and y, just as we confirmed the validity of the pair (2,2) using the values given. Algebraic solutions provide us with a systematic and logical approach to determining the exact values that solve the system of equations.