Simultaneous equations
usually refers to equations of the form
(1)
(2)
We solve these be equating
the coefficients of
or
and
then eliminating that term. For example, in this case we can multiply
(1) by 3 to get
or
and
then eliminating that term. For example, in this case we can multiply
(1) by 3 to get
then
subtract (2)
and
from (1),
Our problem here is to solve
equations such as
(1)
(2)
The general approach is to
rearrange the linear equation (2) to make
the subject say, then substitute the rearranged equation into the
quadratic to find a quadratic equation in
which
we solve to find
then
substitute back into the linear equation to find
For
the above example:
the subject say, then substitute the rearranged equation into the
quadratic to find a quadratic equation in
which
we solve to find
then
substitute back into the linear equation to find
For
the above example:
Replace the
in
with
to
get
in
with
to
get
We expand the brackets and
simplify this expression:
We can factorise and solve
the last expression.
If
we
use (2) to find
and
if
we
use (1) to find
we
use (2) to find
and
if
we
use (1) to find
Example:
(1)
(2)
Make
the subject of (2), then substitute the rearranged equation into (1)
to find a quadratic equation in
which
we solve to find
then
substitute this into the (2) to find
the subject of (2), then substitute the rearranged equation into (1)
to find a quadratic equation in
which
we solve to find
then
substitute this into the (2) to find
Replace the
in
with
to
get
in
with
to
get
We expand the brackets and
simplify this expression:
If
we
use (2) to find
and
if
we
use (1) to find
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