### Video Transcript

Consider the matrix π΄ is equal to
negative 10π₯, π₯ plus three π¦, two π₯ minus π§ and the matrix π΅ is equal to
negative 30, 27, 10. Given that the matrix π΄ is equal
to the matrix π΅, determine the values of π₯, π¦, and π§.

In this question, weβre given two
matrices π΄ and π΅. And we can see that the entries in
matrix π΄ are dependent on the three variables π₯, π¦, and π§. In fact, weβre told that these two
matrices are equal. We need to use this information to
determine the values of π₯, π¦, and π§. Remember, we say that a matrix π΄
is equal to a matrix π΅ if all of the entries in the same row and same column are
equal. In fact, this also tells us that
our matrices must have the same number of rows and columns.

In our case, matrix π΄ is negative
10π₯, π₯ plus three π¦, two π₯ minus π§ and matrix π΅ is negative 30, 27, 10. Since weβre told that these two
matrices are equal, entries in the same row and same column must be equal. For example, the entry in the first
row and first column of matrix π΄ is negative 10π₯, and the entry in the first row
and first column of matrix π΅ is negative 30. So for these two matrices to be
equal, these two entries must be equal, so we get negative 10π₯ must be equal to
negative 30. And we can solve this equation for
π₯. We just divide both sides by
negative 10. And this gives us that our value of
π₯ must be equal to three.

Letβs now move on to row one and
column two of our two matrices. In matrix π΄, we can see that the
entry in row one column two is equal to π₯ plus three π¦. And in matrix π΅, the entry in row
one column two is equal to 27. So because these two matrices are
equal, these two entries must be equal. This gives us that π₯ plus three π¦
must be equal to 27. Remember, we already showed that
our value of π₯ must be equal to three, so we can substitute this into our
equation. This gives us that three plus three
π¦ must be equal to 27. We can then solve this equation for
π¦. Weβll start by subtracting three
from both sides of the equation. This gives us that three π¦ is
equal to 24. Now to solve this equation for π¦,
weβll divide both sides of the equation through by three. And so we get that π¦ is equal to
24 divided by three, which is, of course, equal to eight.

We can do the same with the final
entry in each of our two matrices, the entry in row one column three. In matrix π΄, this entry is two π₯
minus π§. And in matrix π΅, this entry is
10. And remember, weβre told matrix π΄
is equal to matrix π΅, so we must have these two entries equal. So we get two π₯ minus π§ is equal
to 10. Remember, we already showed earlier
that if our two matrices are equal, π₯ must be equal to three. So to help us find our value of π§,
weβll substitute π₯ is equal to three into this equation. Substituting π₯ is equal to three,
we get two times three minus π§ is equal to 10. And we can just solve this equation
for π§. We add π§ to both sides and then
subtract 10 from both sides of the equation. This gives us that π§ is equal to
negative four. So this gives us that π₯ is equal
to three, π¦ is equal to eight, and π§ is equal to negative four.

But remember, it can be very useful
to substitute these values back into our matrix to check that our answer is
correct. So letβs substitute π₯ is equal to
three, π¦ is equal to eight, and π§ is equal to negative four into our matrix
π΄. Substituting these values in, we
get that our matrix π΄ is equal to negative 10 times three, three plus three times
eight, two times three minus negative four. And if we evaluate each of these
entries, we see we get negative 30, 27, 10. And each of these entries is
exactly the same as we have in matrix π΅, so we know we have the right answer.

Therefore, if the matrix π΄ is
equal to negative 10π₯, π₯ plus three π¦, two π₯ minus π§ and the matrix π΅ is equal
to negative 30, 27, 10, then for π΄ to be equal to π΅, we must have that π₯ is equal
to three, π¦ is equal to eight, and π§ is equal to negative four.