### Video Transcript

In this video, we will learn how to
solve systems of linear equations using elimination.

Linear equations are equations in
which the highest power of each variable that appears is one. And we also donโt have any terms in
which different variables are multiplied together. For example, the equation two ๐ฅ
plus three ๐ฆ equals seven is a linear equation. It involves two variables or
unknowns, ๐ฅ and ๐ฆ. And as it involves two variables,
we canโt solve this equation on its own; we need more information.

If we were given a second linear
equation also involving these same two variables such as the equation five ๐ฅ minus
๐ฆ equals nine, then we now have whatโs called a system of linear equations. In general, we need the same number
of equations as there are variables. So in this system, we have two
variables ๐ฅ and ๐ฆ and two equations. So weโd be able to solve it in
order to find the values of ๐ฅ and ๐ฆ that work in both equations. There are a range of methods that
we can use to do this. In this video, weโre going to focus
on the elimination method. Letโs look then at a series of
different examples, beginning with this one.

Using elimination, solve the
simultaneous equations three ๐ฅ plus two ๐ฆ equals 14, six ๐ฅ minus two ๐ฆ
equals 22.

Now, simultaneous equations are
just another way of saying a system of equations. We can see that we have two
equations. And theyโre each in the same
two variables, ๐ฅ and ๐ฆ. Weโre told that we need to use
the method of elimination to answer this question. So letโs see what this looks
like. The principle of this method is
that we can eliminate or get rid of one of the two variables from our two
equations. We can choose to eliminate
either ๐ฅ or ๐ฆ. But to make things easier, we
notice that in this question, we have two ๐ฆ in each equation. But in equation 1, two ๐ฆ is
being added to the ๐ฅ-term and in equation 2 itโs being subtracted from the
๐ฅ-term.

The key thing we need to spot
is that if we were to add these two entire equations together, then weโll
eliminate the ๐ฆ-term. Letโs see what that looks
like. On the left-hand side, three ๐ฅ
plus six ๐ฅ gives nine ๐ฅ. We then have positive two ๐ฆ
plus negative two ๐ฆ. Thatโs two ๐ฆ minus two ๐ฆ,
which is equal to zero. On the right-hand side, we have
14 plus 22, which is equal to 36. So weโve eliminated the
๐ฆ-variables and created an equation in ๐ฅ only. Nine ๐ฅ is equal to 36.

Now that our equation is in
terms of ๐ฅ only, itโs straightforward to solve to find the value of ๐ฅ. We have nine ๐ฅ equals 36, so
we need to divide both sides of the equation by nine. Doing so gives ๐ฅ equals
four. So weโve found the value of one
of our two variables. Next, we need to find the value
of our ๐ฆ-variable. And to do this, we can
substitute the value weโve found for ๐ฅ into either of our two equations. It really doesnโt matter which
we choose. Iโm going to choose to use
equation 1 simply because the coefficient of ๐ฆ is positive in this equation, so
it will make things a little easier.

So substituting ๐ฅ equals four
gives three times four plus two ๐ฆ is equal to 14. Three times four is, of course,
equal to 12. So we have the equation 12 plus
two ๐ฆ equals 14, which is an equation in ๐ฆ only. To solve, we first need to
subtract 12 from each side to give two ๐ฆ is equal to two and then divide each
side of the equation by two to give ๐ฆ is equal to one. So weโve also found the value
of ๐ฆ. And therefore, weโve solved the
simultaneous equations. Our solution is a pair of
values: ๐ฅ is equal to four and ๐ฆ is equal to one.

Now itโs always a good idea to
check our answer where we can. And in order to do this, weโre
going to substitute the pair of values we found for ๐ฅ and ๐ฆ into whichever
equation we didnโt use when determining the second value. So weโre going to substitute
into equation 2. Substituting ๐ฅ equals four and
๐ฆ equals one on the left-hand side gives six multiplied by four minus two
multiplied by one. Thatโs 24 minus two, which is
equal to 22. And that is indeed the value
that we should have on the right-hand side of the equation. So this confirms that our
solution is correct.

The key principle of the method
of elimination in this question then was to notice that we had almost the same
coefficient of ๐ฆ in each equation, but one was positive and one was
negative. We, therefore, found that if we
were to add the two equations together, this would eliminate the ๐ฆ-variables,
leaving an equation in ๐ฅ only. Our solution is ๐ฅ equals four
and ๐ฆ equals one.

In our next example, weโll see how
the method of elimination works if we canโt eliminate one variable by adding the two
equations together.

Use the elimination method to
solve the simultaneous equations three ๐ plus two ๐ equals 14, four ๐ plus
two ๐ equals 16.

So we have a pair of
simultaneous equations or a system of linear equations in two variables ๐ and
๐. And weโre told that we must use
the elimination method in order to solve this system of equations. Weโll label our two equations
as equation 1 and equation 2 for ease of referencing them. And looking at the two
equations, we notice, first of all, that they have exactly the same coefficient
of ๐. They both have positive two
๐. Now your first thought may be
that we can, therefore, eliminate the ๐-variable by adding the two equations
together. But letโs see what that looks
like.

On the left-hand side, three ๐
plus four ๐ gives seven ๐. We then have positive two ๐
plus another positive two ๐, which gives positive four ๐. And on the right-hand side, we
have 14 plus 16, which is equal to 30. So we have the equation seven
๐ plus four ๐ equals 30. This equation still involves
both variables, so we havenโt achieved our aim of eliminating one, which means
that adding the two equations together wasnโt the correct step to take.

Instead, letโs try subtracting
one equation from the other. And as the coefficient of the
other valuable, ๐, is greater in equation 2 than it is in equation 1, Iโm going
to try subtracting equation 1 from equation 2. On the left-hand side, four ๐
minus three ๐ gives ๐. We then have two ๐ minus two
๐. So that cancels out to
zero. And on the right-hand side, 16
minus 14 is two. So we have ๐ equals two. Weโve eliminated the
๐-variable. And in fact, we found the
solution for ๐ at the same time. The correct way to eliminate
one variable then was to subtract one equation from the other. And the reason for this is that
the coefficients of the variable we were trying to eliminate, that is the ๐โs,
are identical in both equations and they have the same sign.

There is a helpful acronym that
we can use to help us remember this, SSS. It stands for if we have the
same signs, then we subtract. We must remember that it is the
signs of the variable we are looking to eliminate that is important. So itโs the ๐โs that we were
interested in here. As the signs of the ๐โs were
the same, we eliminated them by subtracting one equation from the other.

Now that we found the value of
๐, we need to find the value of ๐, which we can do by substituting our value
of ๐ into either of the two equations. Letโs choose equation one. We have three multiplied by two
plus two ๐ is equal to 14. Thatโs six plus two ๐ equals
14. And subtracting six from each
side gives two ๐ is equal to eight. We then solve for ๐ by
dividing each side of the equation by two, giving ๐ equals four.

So we have our solution to the
simultaneous equations: ๐ is equal to two and ๐ is equal to four. But we should check our answer,
which we can do by substituting the pair of values we found into the other
equation. Thatโs equation 2. Substituting ๐ equals two and
๐ equals four into the left-hand side of equation 2 gives four times two plus
two times four. Thatโs eight plus eight, which
is equal to 16, the value on the right-hand side of equation 2. So this confirms that our
solution is correct.

We need to remember then that
helpful acronym SSS, which stands for if the signs of the variable we want to
eliminate are the same, then we subtract. Of course, the reverse is also
true. If the signs of the variable we
want to eliminate are different, then we add. Our solution to this set of
simultaneous equations which weโve checked is ๐ equals two and ๐ equals
four.

Now, in the two examples weโve seen
so far, weโve been able to eliminate one of the variables straightaway by adding or
subtracting the original two equations. Sometimes, though, there may be an
extra step needed before we can do this, which weโll see in our next example.

Using elimination, solve
simultaneous equations five ๐ฅ minus four ๐ฆ equals 21, four ๐ฅ plus 12๐ฆ equals
32.

So weโre asked to solve this
system of equations using the elimination method, which means weโre looking to
eliminate either the ๐ฅ- or ๐ฆ-variable by adding or subtracting our two
equations. However, if we were to try this
as the equations currently are, weโd find that in both cases we still have ๐ฅ-
and ๐ฆ-variables in the equation we are left with. If we were to add, weโd have
the equation nine ๐ฅ plus eight ๐ฆ equals 53. And if we were to subtract,
weโd have the equation ๐ฅ minus 16๐ฆ equals negative 11. So this hasnโt actually
helped.

So why hasnโt it worked? Well, in order to use the
method of elimination, weโre looking for the coefficients of one of the
variables to be the same in both equations, or at least to have the same
magnitude such as positive and negative three. But in this problem, this isnโt
the case. We have a coefficient of five
for ๐ฅ in the first equation and four in the second. And we have a coefficient of
negative four for ๐ฆ in the first equation and 12 in the second. So simply adding or subtracting
these equations as they currently are doesnโt eliminate either variable.

Weโre told, though, that we
need to use the method of elimination. So what should we do? Well, what weโre going to do is
manipulate these equations slightly so that we do have the same coefficient, or
at least the same magnitude of coefficient for one of the variables. To achieve this, we use the
equality property of multiplication, which says that if we multiply both sides
of an equation by the same value, then the equality is still true. Weโre looking for a value that
we can multiply one equation by which will then give the same coefficient, or
same-size coefficient, for one of the variables in both equations.

Looking at the ๐ฆ-variables in
our two equations, we see that four is a factor of 12. So if we were to multiply
equation 1 by three, then we would have negative 12๐ฆ. And so the magnitude of the
coefficients of ๐ฆ would be the same in both equations. Letโs try that then. Letโs multiply equation 1 by
three. So we multiply everything in
equation 1 by three, giving 15๐ฅ minus 12๐ฆ is equal to 63. The coefficients of ๐ฆ in our
two equations are now the same size but with different signs, which means we can
eliminate the ๐ฆ- variables by adding our two equations together.

When we do, we have 15๐ฅ plus
four ๐ฅ, which gives 19๐ฅ; negative 12๐ฆ plus 12๐ฆ, which gives zero; and on the
right-hand side 63 plus 32, which is 95. So weโve eliminated the
๐ฆ-variable from our equations, giving a single equation in ๐ฅ, which we can
solve by dividing both sides by 19. Doing so gives the solution for
๐ฅ. ๐ฅ is equal to five.

We can then find the value of
๐ฆ by substituting ๐ฅ equals five into any of the three equations, either of the
original two or the equation we created when we multiplied equation 1 by
three. Iโm going to use equation 2 as
all of the coefficients are positive in this equation. Doing so gives four times five,
which is 20, plus 12๐ฆ equals 32. We can then subtract 20 from
each side and divide by 12 to give ๐ฆ equals one. So we have our solution: ๐ฅ
equals five and ๐ฆ equals one. But of course, we should check
it. Iโm going to choose to check by
substituting the values into equation 1. Thatโs five ๐ฅ minus four ๐ฆ
equals 21. Substituting ๐ฅ equals five and
๐ฆ equals one gives five times five minus four times one. Thatโs 25 minus four, which is
indeed equal to 21. So this confirms that our
solution is correct.

The key stage in this question,
then, was to notice that we couldnโt eliminate either variable by adding or
subtracting the two equations in their original form. Instead, we had to first multiply
one equation by a constant to create an equivalent equation in which the coefficient
of ๐ฆ was the same magnitude as it was in the second equation. Only then could we use the method
of elimination to solve these simultaneous equations.

Now it may not always be possible
to just multiply one equation by a constant. Instead, we may need to multiply
both equations by different constants. Letโs have a look at an example of
this.

Using elimination, solve the
simultaneous equations four ๐ฅ plus six ๐ฆ equals 40 and three ๐ฅ plus seven ๐ฆ
equals 40.

In the two equations weโve been
given, the coefficients of ๐ฅ are different and the coefficients of ๐ฆ are also
different, which means we canโt eliminate one variable by just adding or
subtracting the equations together. We also notice that neither of
the coefficients of ๐ฅ are factors of the other and neither of the coefficients
of ๐ฆ are factors of each other. We want to create equations in
which the coefficients of either variable are the same or at least the same but
with different signs. So how are we going to do
this?

Well, weโre going to have to
multiply both equations by some constant. Iโm going to choose to multiply
equation 1 by three and equation 2 by four because this will create 12๐ฅ in each
equation. We could also have chosen to
multiply equation 1 by seven and equation 2 by six as this would create 42๐ฆ in
each equation. It doesnโt matter which
variable we choose to eliminate.

Now that we have our two new
equations, we have 12๐ฅ in each. And as the signs are the same,
we can eliminate the ๐ฅ-variable by subtracting. Iโm actually going to subtract
the top equation from the bottom one because the coefficient of ๐ฆ is greater in
the second equation. We have 12๐ฅ minus 12๐ฅ, which
cancels out; 28๐ฆ minus 18๐ฆ, which gives 10๐ฆ; and 160 minus 120, which is
40. So weโve eliminated the
๐ฅ-variable from our equation. We can then solve for ๐ฆ by
dividing both sides of this equation by 10, giving ๐ฆ equals four.

To solve for ๐ฅ, we substitute
this value of ๐ฆ into any of our four equations. Iโm going to choose equation
1. Doing so gives a
straightforward linear equation for ๐ฅ which we can solve by subtracting 24 and
dividing by four to give ๐ฅ equals four. So we have our solution. Both ๐ฅ and ๐ฆ are equal to
four. As always, we should check our
solution by substituting our values into any of the other equations. Iโve used equation 2. And it confirms that our
solution is correct.

The key step in this question
then was to multiply both equations by some number to create the same
coefficient of one of the variables. We could then use the method of
elimination to eliminate this variable and solve our simultaneous equations.

Letโs review some of the key points
from this lesson. Our aim in this method is to
eliminate one variable by adding or subtracting the two equations. If the coefficients of the variable
we wish to eliminate have different signs, we add the two equations together. And if the signs are the same, then
we subtract the equations, which we can remember using the acronym SSS. We also saw that in some cases we
may need to multiply one or both equations by a constant before we can add or
subtract to eliminate one variable.

Once weโve eliminated one variable
and found the value of the other, we need to substitute this value back into one of
our equations in order to find the value of the variable we eliminated. And we should always check our
answer by substituting both values into whichever equation we didnโt use when
working out the second value.