Video Transcript
Using elimination, solve the
simultaneous equations three 𝑥 plus two 𝑦 equals 14, six 𝑥 minus two 𝑦
equals 22.
Now, simultaneous equations are
just another way of saying a system of equations. We can see that we have two
equations. And they’re each in the same
two variables, 𝑥 and 𝑦. We’re told that we need to use
the method of elimination to answer this question. So let’s 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 𝑥 or 𝑦. But to make things easier, we
notice that in this question, we have two 𝑦 in each equation. But in equation 1, two 𝑦 is
being added to the 𝑥-term and in equation 2 it’s being subtracted from the
𝑥-term.
The key thing we need to spot
is that if we were to add these two entire equations together, then we’ll
eliminate the 𝑦-term. Let’s see what that looks
like. On the left-hand side, three 𝑥
plus six 𝑥 gives nine 𝑥. We then have positive two 𝑦
plus negative two 𝑦. That’s two 𝑦 minus two 𝑦,
which is equal to zero. On the right-hand side, we have
14 plus 22, which is equal to 36. So we’ve eliminated the
𝑦-variables and created an equation in 𝑥 only. Nine 𝑥 is equal to 36.
Now that our equation is in
terms of 𝑥 only, it’s straightforward to solve to find the value of 𝑥. We have nine 𝑥 equals 36, so
we need to divide both sides of the equation by nine. Doing so gives 𝑥 equals
four. So we’ve found the value of one
of our two variables. Next, we need to find the value
of our 𝑦-variable. And to do this, we can
substitute the value we’ve found for 𝑥 into either of our two equations. It really doesn’t matter which
we choose. I’m going to choose to use
equation 1 simply because the coefficient of 𝑦 is positive in this equation, so
it will make things a little easier.
So substituting 𝑥 equals four
gives three times four plus two 𝑦 is equal to 14. Three times four is, of course,
equal to 12. So we have the equation 12 plus
two 𝑦 equals 14, which is an equation in 𝑦 only. To solve, we first need to
subtract 12 from each side to give two 𝑦 is equal to two and then divide each
side of the equation by two to give 𝑦 is equal to one. So we’ve also found the value
of 𝑦. And therefore, we’ve solved the
simultaneous equations. Our solution is a pair of
values: 𝑥 is equal to four and 𝑦 is equal to one.
Now it’s always a good idea to
check our answer where we can. And in order to do this, we’re
going to substitute the pair of values we found for 𝑥 and 𝑦 into whichever
equation we didn’t use when determining the second value. So we’re going to substitute
into equation 2. Substituting 𝑥 equals four and
𝑦 equals one on the left-hand side gives six multiplied by four minus two
multiplied by one. That’s 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 𝑦 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 𝑦-variables,
leaving an equation in 𝑥 only. Our solution is 𝑥 equals four
and 𝑦 equals one.
