Question Video: Solving a System of Two Equations Using Determinants Mathematics

Use determinants to solve the system −8𝑥 − 4𝑦 = −8, 9𝑥 − 6𝑦 = −9.

03:51

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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.