Three types of equation systems exist in mathematics: inconsistent, dependent and independent. An inconsistent system is fairly simple to recognize, given they graph as parallel lines. Dependent and independent systems, however, may be a little trickier to remember or easier to confuse. The two methods for discovering what type of system an equation (or set of them) might be are via graphing them out, or simply solving the equations. Either way will result in the proper solution.
Look at the assignment. Note any special instructions given, such as "graph these equations" or "solve for x" that may require more than a calculator or paper and pencil, and then continue.
When asked to graph the set of equations given, use the calculator or the graph paper to do so. If asked to solve the equations for x and y, write them out correctly and do so.
Example Set: y = x + 5 and 2y = 2x + 10.
Look at the graphed lines or the solution. If the graphed lines are the same line, then the equation is dependent. If the solution winds up being 10 = 10, which is "true," then the system is dependent. This means that every point in each equation crosses one another from here to forever. In other words, there are an infinite number of solutions and they have the same slope and intercept. Look at the graphed lines. If the slopes are different, it's independent. If they also have different intercepts with those different slopes, they are independent.
Look at the graphed lines or the solution. If the graphed lines are not parallel or a single line, they will ultimately cross at some point. The equations are independent. If the solution winds up being something like x = 6, and then placing it back through results in finding y = 4, then the coordinates where the two lines cross will be (6, 4). As this is the one and only solution, the equations are independent.
Tips & Warnings
- A good way to remember the difference between dependent and independent systems of equations is to consider this: an independent person stands alone, and only one solution exists for an independent equation.
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