Mathematics (Discussion Question week 7) Systems of equations can be solved by graphing, using substitution, or elimination. What are the pros and cons of each method? Which method do you like best? Why? Systems of equations can be solved either by using the method of elimination, by substitution or through graphing or by a combination of more than one of these methods. In the process of elimination, the terms appearing in the equation are rearranged so as to gather like terms together, i.e. those in which the same unknown constant or variable appears (to the same power).
This shows which terms can then be combined to produce a single simplified term or else eliminated altogether, and it should make the equation easier to solve. In the process of substitution, another variable is temporarily introduced into the equation. Usually, this is to represent a complex expression or to help transform the equation into another type, again to make it easier to solve. These two methods of elimination and substitution are algebraic methods. Alternatively, the equation, which may still need to be simplified first, can be graphed to enable the required values to be read or determined visually. Elimination is usually the simplest and quickest of the three methods and it suffices in many cases to solve any system of equation.
In some complex cases however, it may not be clear whether elimination can even be made or not and this process alone becomes insufficient to solve the equations. Substitution can be seen as a relatively more complex procedure because of the use of an additional variable. This method requires extra decision making and steps to not only equate the new variable with an expression involving one or more of the original variables, but also to reconvert back to find the values of the original variables.
Besides, it would be unnecessary if the equation is simple and for which the elimination method can be used instead. However, it has the potential advantage of being a convenient method to quickly make the system of equations appear to be easier to handle in which the new substituted variable(s) has been used. Graphical methods make the task easier to understand visually.
Although it may not always be possible to obtain a precise answer depending on the type of equation and its roots, as long as the graph has been constructed correctly, it does enable at least a good estimate to be obtained. This may not be immediately possible with the algebraic methods in which there are no visual cues so it is useful for quickly ensuring the solution is reasonable. In some cases, as with quadratic equations, it can show whether a solution is even possible i. e.
whether the roots do or do not exist. Similarly, simultaneous equations can be checked for whether the lines do actually intercept and thus the approximate values can be determined for the coordinates where they cross, or it can be seen whether they are parallel, in which case it can be seen that no solution exists. No method would be best in all circumstances so the type of equations determines which method should be used. Generally, the complexity of the equations is the main factor. Personally, I like elimination for its ability to simplify, I see substitution as a useful technique in those cases where it can be applied effectively, and graphing as useful in cases where a visual depiction can help in solving the equations.