What are the differences between graphing, elimination, and substitution?

Graphing, elimination, and substitution are all methods to solve systems of equations, and one difference between them is that elimination only works if all the equations are linear.

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If you are given a system of equations (a set of equations with common solutions), there are a number of general techniques you can use to solve for the solution(s).

If the system of equations consists of only linear equations (all variables appear to the first power only), you can solve the system by guess and check, substitution, linear combinations (elimination), graphing, Cramer's method, Gaussian elimination, inverse matrices, and others.

1. If the system is linear with two equations and two unknowns, we can solve by graphing. We graph the given lines, and the point of intersection (if it exists) is the solution. (If the lines are parallel, there is no solution, and if the lines coincide, there are an infinite number of solutions.)

Graphing does not work well with three equations in three unknowns, as we must graph in three dimensions. Clearly, higher-order systems become impossible (or at least extremely difficult) to visualize, let alone graph.

Note that if the system is not linear, we can still graph the equations, and the point(s) of intersection are the solutions.

2. We can also use substitution. Here, we choose one of the equations and solve for one of the variables. For example, if 2x+y=5, then we can solve for y to get y=-2x+5; or if 2x+3y+z=4, we can solve for z to get z=-2x-3y+4.

We then use this expression for the variable and replace all other occurrences with the expression. If the original system was 2x+y=5 and 3x-2y=7, we can substitute 2x+5 for y in the second equation to get 3x-2(2x+5)=7, yielding a single equation in one unknown.

Each time we use substitution, the number of equations reduces by 1. This method can work even if the equations of the system aren't linear.

3. We can also use elimination. This is also called linear combinations, or the multiplication and addition method. The idea here is to eliminate one of the variables by adding multiples of the equations to each other. This only works if all of the equations are linear.

Last Updated by eNotes Editorial on March 1, 2021
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