Video Transcript
Use determinants to solve the
system negative eight 𝑥 minus four 𝑦 equals negative eight and nine 𝑥 minus six
𝑦 equals negative nine.
So the first thing we want to do
with this problem is actually set up a matrix equation of our system of
equations. And when we do that, what we’re
gonna have is the matrix negative eight, negative four, nine, negative six
multiplied by the matrix 𝑥, 𝑦 is equal to the answer matrix and this is negative
eight, negative nine. So now what we’re gonna do, because
we want to use determinants to solve the system, then what we’re gonna do is use
Cramer’s rule. And Cramer’s rule tells us that 𝑥
is equal to the determinant of the matrix Δ sub 𝑥 over the determinant of the
matrix Δ. And then to find 𝑦, it’s equal to
the determinant of the matrix Δ sub 𝑦 over the determinant of the matrix Δ.
But we might look at this and
think, “Well, what is the matrix Δ sub 𝑥?” Well, in fact, what this is is the
matrix that’s formed when we substitute the answer matrix for the column of
𝑥-coefficients in our original matrix. So, for example, in our problem, we
swap the first column in our matrix for the answer matrix. So instead of reading negative
eight and then nine, it read negative eight and then negative nine. So now let’s move straight on and
find our determinants. So first of all, we want the
determinant of the matrix negative eight, negative four, nine, negative six. So when we work this out, we’re
gonna get negative eight multiplied by negative six minus negative four multiplied
by nine, which is gonna give us 48 plus 36. And this is equal to 84.
And a good thing about this is that
it also helps us to check that we can solve our system of equations because if our
matrix was singular, then the determinant will be equal to zero. So we can see that it isn’t the
case in this problem here. So next, what we’re gonna have a
look at is the determinant of the matrix Δ sub 𝑥. Well, what we already said here is
what Δ sub 𝑥 is, is the matrix we get when we substitute in negative eight and
negative nine, our answer matrix, instead of our 𝑥-coefficients. So we’re gonna get the matrix
negative eight, negative four, negative nine, negative six. So for this determinant, what we’re
gonna get is negative eight multiplied by negative six minus negative four
multiplied by negative nine, which is gonna be equal to 12. Okay, great. So one more determinant to work
out.
So now what we’re looking for is
the determinant of the matrix Δ sub 𝑦. What this is gonna be equal to is
the determinant of the matrix negative eight, negative eight, nine, negative
nine. And as is before, what we’ve got
this by is substituting in our answer matrix for the coefficients of 𝑦. So this is gonna be equal to
negative eight multiplied by negative nine minus negative eight multiplied by nine,
which is gonna be equal to 144. Okay, great. So now we’ve got everything we need
to use Cramer’s rule to solve our system of equations. So, using Cramer’s rule, what we’re
gonna get is 𝑥 is equal to 12 over 84. But then what we can do is divide
the numerator and denominator by 12. And when we do that, we get 𝑥 is
equal to one over seven.
Okay, great. We found the solution for 𝑥. Now let’s move on to 𝑦. So once again, using Cramer’s rule,
we’re gonna get 𝑦 is equal to and we’ve got the determinant of the matrix Δ sub 𝑦
over the determinant of the matrix Δ, so this is gonna give us 144 over 84. So once again, what we can do is
simplify our fraction. We could do that by dividing the
numerator and denominator both by 12. And when we do that, we get 𝑦 is
equal to 12 over seven or twelve-sevenths. So therefore, we can say the
solutions to our equation are 𝑥 equals a seventh and 𝑦 equals twelve-sevenths.
