Video Transcript
Use determinants to solve the system negative nine 𝑥 equals negative eight add eight 𝑦, six 𝑦 equals seven add three 𝑥.
There are lots of methods for solving a system of linear equations. But when we’re asked to use determinants, that’s when we use Cramer’s rule. Cramer’s rule involves converting our system of linear equations into a matrix equation. Recall that Cramer’s rule is the following. We can find 𝑥 by calculating Δ sub 𝑥 over Δ and 𝑦 by calculating Δ sub 𝑦 over Δ. Here, Δ is the determinant of the coefficient matrix, and Δ𝑥 and Δ𝑦 are the determinants of the matrices found by substituting elements of the constants matrix with the elements from the columns of the 𝑥- and 𝑦-coefficients.
So let’s begin this question by converting this system into a matrix equation. Recall that when we put a system like this into a matrix equation, there are three parts. We have the coefficient matrix, the variable matrix, and the constant matrix. In order to put this into matrix form, the first thing we need to do is rearrange our equations into a form that can easily be converted into a matrix equation. We should try to align the 𝑥’s and the 𝑦’s and the constants.
For the first equation, we could add nine 𝑥 to both sides and then add eight to both sides. That gives us nine 𝑥 add eight 𝑦 equals eight. So let’s now try and get the second equation into this similar format. We could do this by subtracting six 𝑦 from both sides and seven from both sides. And that gives us three 𝑥 minus six 𝑦 equals negative seven. Rearranging in this way makes it much easier to put it into a matrix equation. These are the coefficients which go into the coefficient matrix. That’s nine, eight, three, and negative six.
Next, we have the variable matrix. This matrix consists of the variables for our system, so that’s going to be 𝑥 and 𝑦. And finally, we have the constant matrix. This just consists of the constants of our system, so that’s going to be eight and negative seven.
So now we’ve set up our matrix equation, we can look at using Cramer’s rule. Let’s begin by finding Δ sub 𝑥 and Δ sub 𝑦. Remember, Δ sub 𝑥 and Δ sub 𝑦 are the determinants of the matrices found as a result of substituting the elements of the constants matrix with the elements from the columns of the 𝑥- and 𝑦-coefficients. So to find Δ sub 𝑥, we consider the coefficients matrix. But what we do is swap out the 𝑥-coefficients in the coefficient matrix, that’s nine and three, with the elements of the constant matrix, that’s eight and negative seven. So Δ sub 𝑥 is the determinant of the matrix eight, eight, negative seven, negative six.
We now need to actually calculate this determinant. So let’s start by recalling how we find the determinant of a two-by-two matrix. To find the determinant of a matrix 𝑎, 𝑏, 𝑐, 𝑑, we subtract the product of the diagonals, that is, 𝑎𝑑 minus 𝑏𝑐. So the determinant of matrix eight, eight, negative seven, negative six is eight multiplied by negative six minus eight multiplied by negative seven, that is, negative 48 minus negative 56. But that’s just negative 48 add 56. And that gives us eight.
So now we need to do the same for Δ sub 𝑦. That’s going to be the determinant of the coefficient matrix, but with the 𝑦-coefficients replaced with the constant matrix, that is, nine, eight, three, negative seven. So we now need to find the determinant of this matrix. Using the same method as we use for the matrix Δ sub 𝑥, this is nine multiplied by negative seven minus eight multiplied by three. And that gives us negative 63 minus 24, which gives us negative 87. So now we’ve found Δ sub 𝑥 and we’ve found Δ sub 𝑦.
But we still need to find the value for Δ. Δ is the determinant of the coefficient matrix. That is the determinant of the matrix nine, eight, three, negative six, that is, nine multiplied by negative six minus eight multiplied by three, which is negative 54 minus 24. And that gives us negative 78. So now we have Δ sub 𝑥, Δ sub 𝑦, and Δ, we can now apply Cramer’s rule. This tells us that we can find the value of 𝑥 by doing Δ sub 𝑥 over Δ. We already found Δ sub 𝑥 to be eight. And we found Δ to be negative 78. Therefore, 𝑥 is eight over negative 78.
We can actually simplify this fraction by dividing both the numerator and denominator by two, which gives us 𝑥 equals negative four over 39. Cramer’s rule also tells us that 𝑦 is equal to Δ𝑦 over Δ. We found Δ sub 𝑦 to be negative 87 and Δ to be negative 78. So 𝑦 is equal to negative 87 over negative 78. As the highest common factor of 78 and 87 is three, we can divide both the numerator and denominator by three. In fact, we can actually divide our numerator and denominator by negative three, which gives us 29 over 26. So that leads us to our final answer: 𝑥 equals negative four over 39 and 𝑦 equals 29 over 26.
