Video Transcript
Which of the following equations is
represented by the given area model? Option A) four 𝑦 minus two 𝑥
times two 𝑦 plus 𝑥 over two minus three equals eight 𝑦 squared minus two 𝑥𝑦
minus 12𝑦 minus 𝑥 squared minus six 𝑥. Option B) four 𝑦 minus two 𝑥
times two 𝑦 plus two over 𝑥 minus three equals eight 𝑦 squared minus four 𝑥𝑦
minus 12𝑦 minus four plus six 𝑥 plus eight 𝑦 over 𝑥. Option C) four 𝑦 minus two 𝑥
times two 𝑦 plus 𝑥 over two minus three equals eight 𝑦 minus two 𝑥𝑦 minus 12𝑦
minus 𝑥 plus six 𝑥. Option D) four 𝑦 minus two 𝑥
times two 𝑦 plus 𝑥 over two minus three equals eight 𝑦 squared minus two 𝑥𝑦
minus 12𝑦 minus 𝑥 squared plus six 𝑥. Option E) four 𝑦 minus two 𝑥
times two 𝑦 plus 𝑥 over two minus three equals eight 𝑦 minus two 𝑥𝑦 minus 12𝑦
minus 𝑥 minus six 𝑥.
So, let’s have a look at the
question and the area model. An area model is a diagram that
allows us to represent the multiplication of terms as a rectangle. We take the values to be multiplied
as the length and the width. And we would find the area of these
two values by multiplication of the so-called length and the width. So, for example, if we wanted to
multiply the terms three and two 𝑥, then the term inside the rectangle would be six
𝑥 since that’s equal to three times two 𝑥.
Let’s have a look at our diagram
and see if we can find any of the missing values. The missing value in our
highlighted cell would be found by multiplying negative two 𝑥 and negative
three. We know that if we multiply
negative two by negative three, this would give us a positive value of six. So, negative two 𝑥 times negative
three will give us six 𝑥, which we can write into our area model. Let’s see what other values we can
work out.
If we consider that in our area
model we’re multiplying two values to give us an answer, this means that in our
model we’re looking for two known values and an unknown, which could be two lengths
and an unknown area or unknown area value and one missing length. So, let’s have a look at the
missing length highlighted along the top. Here, we need to calculate negative
two 𝑥 times what equals negative four 𝑥𝑦. If we look at our coefficients,
then negative two times two must give us negative four. If we look at our 𝑥-values, we
have an 𝑥 on the left-hand side and an 𝑥 on the right-hand side, so we don’t
multiply in anymore.
On the right-hand side, we have a
𝑦 term, so we must multiply by two 𝑦. So, we can add this term to our
area model. The next missing length can be
found since we know that that value times negative three would give us negative
12𝑦. And the answer must be four 𝑦,
since four times negative three would give us negative 12. And we must have a 𝑦 term on the
left-hand side and the right-hand side. Next, we can calculate the missing
area value formed from two 𝑦 times four 𝑦, which is eight 𝑦 squared.
Our next missing length, can be
found by calculating negative two 𝑥 times what will give us negative 𝑥
squared. Considering our coefficients, this
means negative two times what will give us negative one, which means that our
missing coefficient must be a half. Considering our 𝑥 terms then, we
have 𝑥 times what will give us 𝑥 squared, leaving us with 𝑥. And so, our missing term must be a
half 𝑥. We can leave it as a half 𝑥 or we
can write it in the slightly neater format of 𝑥 over two. And so, our final missing value
will be found by calculating four 𝑦 times 𝑥 over two, which will be four 𝑥𝑦 over
two.
We can notice that two will go into
the numerator and the denominator, giving us a simplified answer of two 𝑥𝑦, which
we can fill into the table. So, now, we have all the values in
our area model, we can determine the two original expressions that were being
multiplied. The first expression would be
composed of four 𝑦 and negative two 𝑥, which we write in parentheses as four 𝑦
minus two 𝑥. The second expression is composed
of two 𝑦, 𝑥 over two, and negative three, giving us two 𝑦 plus 𝑥 over two minus
three in parentheses.
We notice that the question has
asked us for an equation and not for an expression. So, we must also evaluate the
result of our expressions being multiplied. And we can do this using our area
model. We need to take each individual
area and add them together to get a total area. This will be eight 𝑦 squared plus
two 𝑥𝑦 minus 12𝑦 minus four 𝑥𝑦 minus 𝑥 squared plus six 𝑥. We notice that we have two terms in
𝑥𝑦, so we can simplify this expression further. And so, now, we have an answer that
matches option D, four 𝑦 minus two 𝑥 times two 𝑦 plus 𝑥 over two minus three
equals eight 𝑦 squared minus two 𝑥𝑦 minus 12𝑦 minus 𝑥 squared plus six 𝑥.
