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The whole problem with solving
a system of equations is that
you cannot solve an equation
that has two unknowns in it.
You need an equation with
only one variable so that
you can isolate the variable
on one side of the equation.
Both methods that we will
look at are techniques for
eliminating one of the variables
to give you an equation in
just one unknown, which you
can then solve by the usual
methods.
The first method of solving
systems of linear equations
is the addition method, in
which the two equations are
added together to eliminate
one of the variables.
Adding the equations means
that we add the left sides
of the two equations together,
and we add the right sides
together. This is legal because
of the Addition Principle,
which says that we can add
the same amount to both sides
of an equation. Since the
left and right sides of any
equation are equal to each
other, we are indeed adding
the same amount to both sides
of an equation.
Consider this simple example:
Example:
If we add these equations
together, the terms containing
y will add up to zero
(2y plus -2y),
and we will get
or
5x = 5
x = 1
However, we are not finished
yet—we know x, but
we still don’t know y.
We can solve for y
by substituting the now known
value for x into either
of our original equations.
This will produce an equation
that can be solved for y:
Now that we know both x
and y, we can say that
the solution to the system
is the pair (1, 1/2).
This last example was easy
to see because of the fortunate
presence of both a positive
and a negative 2y.
One is not always this lucky.
Consider
Example:
Now there is nothing so obvious,
but there is still something
we can do. If we multiply
the first equation by -3,
we get
(Don’t forget to multiply
every term in the equation,
on both sides of the equal
sign). Now if we add them
together the terms containing
x will cancel:
or
As in the previous example,
now that we know y
we can solve for x
by substituting into either
original equation. The first
equation looks like the easiest
to solve for x, so
we will use it:
And so the solution point
is (-4, 7/2).
Now we look at an even less
obvious example:
Example:
Here there is nothing particularly
attractive about going after
either the x or the
y. In either case,
both equations will have to
be multiplied by some factor
to arrive at a common coefficient.
This is very much like the
situation you face trying
to find a least common denominator
for adding fractions, except
that here we call it a Least
Common Multiple (LCM). As
a general rule, it is easiest
to eliminate the variable
with the smallest LCM. In
this case that would be the
y, because the LCM
of 2 and 3 is 6. If we wanted
to eliminate the x
we would have to use an LCM
of 10 (5 times 2). So,
we choose to make the coefficients
of y into plus and
minus 6. To do this, the first
equation must be multiplied
by 3, and the second equation
by 2:
or
Now adding these two together
will eliminate the terms containing
y:
or
x = 2
We still need to substitute
this value into one of the
original equation to solve
for y:
Thus the solution is the
point (2, 2). |
