A system of linear equations involves two relationships with two variables in each relationship. By solving a system, you are finding where the two relationships are true at the same time, in other words, the point where the two lines cross. Methods for solving systems include substitution, elimination, and graphing. Each one will give the right answer but is more or less useful depending on the problem and situation.
This method involves plugging an expression from one equation in for the variable in another. To use this method, at least one variable in one of the equations must be isolated. This is why substitution is most useful when the problem already contains an isolated variable or if there is at least a variable that has a coefficient of one. If you can solve basic algebra equations very quickly, substitution is a good choice. However, it poses problems for those who tend to make arithmetic mistakes.
To use elimination, you must line up both equations vertically with the variables on one side and constants on the other. The bottom equation is then subtracted from the top one to cancel out a variable. This makes elimination efficient when the constants of both equations are already isolated. Additionally, if the coefficients of the Xs or Ys in both equations are the same, elimination will get a solution quickly with minimal steps. On the other hand, sometimes one or both whole equations have to be multiplied by a number to make the variable cancel. This can make the work take longer, and elimination is not the best choice in this scenario.
Graphing by Hand
If the equations do not involve fractions or decimals, and you have a good visual understanding of linear equations, graphing on the coordinate plane is a good option. This technique involves visually finding the point on the graph where the two lines cross to get the solutions for X and Y. Because it helps you to graph quickly, having both equations in Y= form makes this method useful. In contrast, if neither equation has Y isolated, you are better off using substitution or elimination.
Graphing on a Calculator
Using a graphing calculator to enter both equations and find the point of intersection comes in handy when they involve decimals or fractions. It is also a good choice when the teacher allows such calculators on tests or quizzes. However, as in graphing by hand, this technique works best when the Ys in both equations are already isolated.
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