### Video Transcript

In a shop, four T-shirts and two
jumpers cost 81 pounds 80 and three T-shirts and three jumpers cost 92 pounds
10. Work out the individual cost of a
T-shirt and a jumper.

To answer this question, we’re
going to need to use algebra. And we’re going to need to
translate the information we’ve been given into a pair of equations. First, we’ll introduce some
letters. And it doesn’t really matter which
letters we use. But perhaps, in this question, it
makes sense to use 𝑡 to represent the cost of one T-shirt and 𝑗 to represent the
cost of one jumper.

Let’s look at the first statement
we’re given. Four T-shirts, so that’s four 𝑡,
and two jumpers, so that’s plus two 𝑗, cost 81 pounds 80 which we can write as
81.8. So this gives us our first
equation. Now, let’s look at the second
statement we’re given. Three T-shirts — so that’s three 𝑡
— and three jumpers — plus three 𝑗 — cost 92 pounds 10 which you can write as
92.1. So now, we have our second
equation.

What we’ve now created are a pair
of linear simultaneous equations in 𝑡 and 𝑗, which we need to solve to find each
of their values. We’ll answer this question using
the elimination method. And what we want to do first of all
is manipulate each of these equations. That’s just multiplying or divide
them by some number so that we end up either the same number of 𝑡 or the same
number of 𝑗 in the two equations.

There are various different ways
that we could do this. But let’s consider equation one
first. And we can actually divide every
term in equation one by two. Four 𝑡 divided by two gives two
𝑡. Two 𝑗 divided by two just gives
one 𝑗 or 𝑗. And 81.8 divided by two gives
40.9. This equation is equivalent to
equation one because we’ve done the same thing to every term. But it’s slightly simpler because
the coefficient of 𝑗 is just one. We can label this equation as
equation three.

Now let’s consider equation
two. We can divide every term in this
equation by three. Three 𝑡 divided by three is
𝑡. Three 𝑗 divided by three is
𝑗. And 92.1 divided by three is
30.7. Again, this equation is exactly
equivalent to equation two as we did the same thing to all three terms. But it’s slightly simpler. And again, we have a coefficient of
𝑗 which is just one which now matches the coefficient of 𝑗 in equation three.

We can now subtract equation four
from equation three. Two 𝑡 minus 𝑡 is one 𝑡 which we
just write as 𝑡. And 𝑗 minus 𝑗 is zero. The terms cancel out. So we’re left with an equation in
𝑡 only. On the right, we have 40.9 minus
30.7 which is 10.2. We found then the value of 𝑡. It’s 10.2.

Remember though that this
represented the cost of a T-shirt. So it needs to be in pounds. And we need two decimal places. So we can add a zero in the second
decimal place. We found that the cost of a T-shirt
is 10 pounds 20.

To find the value of 𝑗, we can
then substitute the value of 𝑡 into any of our four equations. I’m going to choose to substitute
into equation four. Equation four was 𝑡 plus 𝑗 equals
30.7. So now we have 10.2 plus 𝑗 equals
30.7. To solve this equation for 𝑗, we
need to subtract 10.2 from each side. And it gives 𝑗 equals 20.5.

Again, we need to write this as a
value with the pound sign and with a zero in the second decimal place. So we found that the cost of a
jumper is 20 pounds 50.

It’s always good to check our
answers where we can. So we can substitute the values
we’ve found for 𝑡 and 𝑗 back into any of these equations. I’m going to substitute into
equation one. The left-hand side of equation one
was four 𝑡 plus two 𝑗. So substituting 10.2 for 𝑡 and
20.5 for 𝑗, we have four multiplied by 10.2 plus two multiplied by 20.5. This gives 40.8 plus 41 which is
equal to 81.8. This is equal to the value on the
right-hand side of equation one and the same as the cost of four T-shirts and two
jumpers.

So this confirms that our solution
is correct. You could also calculate the cost
of three T-shirts and three jumpers if you wish to again check the answer. We found that the individual cost
of one T-shirt is 10 pounds 20 and the cost of one jumper is 20 pounds 50.