Algebra deals mainly with equations of one variable.

At the end of MAT 117 (or whatever class you took),

you learned how to solve equations with more than

one variable in them , such as:

x + 3y = -5

4x - 2y = 8

How?

The key to solving these equations is elimination ,

where we multiply (both sides of) one equation by

a number, (both sides of) the other equation by a

number, and add the two new equations:

-4 × (x + 3y = -5)

1 × (4x - 2y = 8)

The key to solving these equations is elimination,

where we multiply (both sides of) one equation by

a number, (both sides of) the other equation by a

number, and add the two new equations:

-4x - 12y = 20

4x - 2y = 8

The key to solving these equations is elimination,

where we multiply (both sides of) one equation by

a number, (both sides of) the other equation by a

number, and add the two new equations:

So y = 28/-14 = -2. How do we find x?

We can use elimination to find x , or we can

substitute the value of y into one of the original

equations:

x + 3y = -5

x + 3 • (-2) = -5

x = -5 - 3 • (-2) = 1

This method of finding x is called back substitution.

Going back to the original equations:

x + 3y = -5

4x - 2y = 8

How specifically do we answer the question \What

are the solutions ?"?

Answer: By giving the value of x together with the

value of y, usually as an ordered pair (x, y) = (1,-2)

or as a \vector" or as two equations:

x = 1

y = -2

Okay, now what about three equations with three

variables? Such as:

x - y + 2z = 3

2x - 2y + 8z = 22

x - 2y = 2

It's not clear how to come up with an equation with

just one variable in it. However, elimination can

be used to come up with two equations which only

involve the variables y and z. These equations in

turn can be solved for y and z, which in turn will

give us the value of x.

The bad news is that it takes a lot more work and a

lot more care, but it can be done.

But first, it's time for some vocabulary : : :

A linear equation is an equation of the form

number × variable + number
× variable

+ … + number × variable = number

such as:

2x + 3y + 2z = -1

A system of linear equations is one or more linear

equations, which must all be true at the same time.

A system of linear equations is consistent if it has at

least one solution . (Not all systems are consistent!)

A system of linear equations is inconsistent if it has

no solutions.

A system of linear equations is homogeneous if all the

numbers on the right -hand side are 0's. (You may

have run across this term , if you've taken Differential

Equations.)

Two systems of linear equations are equivalent if they

have the same solutions.