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Solving Systems of Linear Equations

There are a number of ways of accomplishing this task. This is one way that you should
consider using if you do not already have a technique the works reliably. I will
demonstrate this first for a set of two equations in two unknowns and then for three
equations in three unknowns. If you have a larger set the technique works quite well but
the amount of time you will need to spend grows quickly.

Solve for x and y .


The first thing I always do is to write the equations in a consistent form. I like to put the
constant term on the left side and the variables on the right in some sensible order. In this
case I will use alphabetical order – first x then y. I will also number the equations so that
I can refer to them easily.

Now I pick one of the two variables to eliminate. I look for an easy relationship between
the coefficients. In this case the coefficients of the y variable qualify. The idea is to
multiply one ( or both ) equations by some number ( or pair of numbers ) so that the two
y coefficients ( because I picked on y ) are equal and opposite. Then when I add the two
equations
( I can add the same value to both sides of an equation and still have an
equality. ) the resulting coefficient for y will be zero . I will schematically represent the
operation in this way


This says that equation (3) will be the sum of equation (1) and 4 times equation (2). I
will write equation (1) on one line, 4 times equation (2) on the next line, and then the sum
of the two on the third line.

Now add these to get

The last line is equation (3)

I can solve (3) for x by multiplying both sides by .

With this value for x in hand I may select either (1) or (2) and replace x with . I will
choose (2).

And thus we have the values for x and y that satisfy both of our initial equations. Starting
on the next page I will use the same procedure for a system of three equations in three
unknowns. I will not put in all of the commentary or all of the steps . See if you can
supply those for yourself.

Solve this system for values of x, y, and z that satisfy all three equations simultaneously .

Do what first?

I will choose to get rid of z. I will need to end up with two equations in the two
unknowns x and y so I must do the procedure twice. I will combine (1) with (2) and then
I will combine (1) with (3).

Add

or

Repeat for the next pair of equations.

Add

or

Now treat (4) and (5) as a system of two equations in two unknowns.

Add

From this we see that

Combine (6) with (4) to find x.

Combine (6) and (7) with (3) to find z.

Just for fun you might try these values in (1) and (2) to see that they do work.

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