Title: Elimination Method

This method is useful for any number of equations in any number of unknowns. It essentially consists of eliminating the variables from the equations one by one, until the system looks like an upside-down staircase.

Example:

Consider the following system of 3 equations in 3 unknowns:

x + y = 2

2x + 3y + z = 4

x + 2y + 2z = 6

Our goal is to transform this system into an equivalent system from which it is easy to find the solutions. We now do this step by step.

Subtract 2*(Row1) from Row2 and place the result in the second row;

Subtract Row1 from Row2 and place in the third row. Leave Row1 as is.

x + y = 2

y + z = 0

y + 2z = 4

Subtract Row2 from Row3, and place the result in row3.

Leave Row1 and Row2 as they are.

x + y = 2

y + z = 0

z = 4

From the last form of the system we can deduce the following unique solution to the system:

z = 4

y = -4

x = 2-(-4) = 6

Equivalently, we say that the unique solution to this system is (x, y, z) = (6, -4, 4).

Comments :

We have placed the variables in columns. This makes the system easier to work with now, and prepares us for work with matrices in coming sections.

Not all the systems are so nice and neat; after going through this process you might end up with fewer equations than variables, or equations that contradict each other. In the first case you have a system withinfinitely many solutions; in the second, the system isinconsistent and so has no solutions at all. Go to these pages for a discussion of these situations and the solutions that arise through them.

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