Use determinants to solve the system.
Given a system of linear equations, Cramer’s rule is a handy way to solve for just one of the variables without having to solve the whole system of equations. However, in this case they want us to find all of them: 𝑥, 𝑦, and 𝑧.
So let’s first begin by taking our constants and moving them to the right-hand side. This way we can make this in terms of a matrix equation. Now that the constants are isolated, over on the right-hand side of the equation, we can turn this into a matrix equation.
So first let’s take all of the coefficients and put it into a matrix. Next, we need to take this matrix and multiply it by the 𝑥𝑦𝑧 matrix. Our answer column which is one, negative three, negative one. Cramer’s rule states that we can solve for 𝑥, 𝑦, and 𝑧 using this formula. So what are all these symbols meaning? The triangle itself is the matrix of coefficients which is located here. So this triangle 𝑥 means it’s the matrix where the 𝑥 column will be replaced with the answer column, same thing with the 𝑦 and the 𝑧. So lastly, what are all of these lines that look like absolute value lines. Those represent, to take the determinant. So we will need to take the determinate of each of these matrices.
So let’s begin with finding the determinant of the coefficient matrix, which is located on the denominator of every single fraction. So the determinant of this big three-by-three matrix will begin by taking the top left-hand corner number, three. And multiplying by, and we multiply by the determinant of the numbers that are not in the row or column of the three that we began with. And then we subtract the top middle number, two, times the determinant of all of the numbers that are not in the row or column of the two. And then we add negative two times the determinant of these numbers, the numbers not- that are not in the row or column of the negative two. So we take three times this determinant.
So how do we find a determinant? We multiply the numbers that are diagonal and then subtract the other numbers that are diagonal, and now we repeat. So now we need to subtract two times, begin with the top left-hand corner number, three, and negative five, we multiply and then we subtract. And after subtracting negative two times negative three, now we take negative two and multiply by this determinant, three times four minus negative two times three. And now we simplify. So we have three times negative three minus two times negative 21 minus two times 18, and we get negative three which is the answer to the determinant of the matrix coefficient. So we can replace all of the dominators with negative three.
So now we take the coefficient matrix, except where the 𝑥 column is, we replace it with the answer column. And now we repeat our steps to find the determinant of a three-by-three matrix. So we take one times the determinant of this matrix minus two times the determinant of these numbers plus negative two times the determinant of these numbers. So now we start evaluating just like we did before. And after simplifying, we get an answer of negative nine. So when solving for 𝑥, negative nine divided by negative three means 𝑥 is equal to three.
So now let’s do the exact same process but with 𝑦. So we take the coefficient matrix but now instead of the 𝑦 column, replace it with the answer column. So we have three times the determinant of these numbers minus one times the determinant of these numbers plus negative two times the determinant of these numbers. So now we evaluate the determinants and then we multiply and simplify, and we get 75. And 75 divided by negative three means 𝑦 is negative 25.
So lastly, 𝑧. Let’s go ahead and replace the 𝑧 column with the answer column. And now we evaluate. Three times the determinant of these numbers minus two times the determinant of these numbers plus one times the determinant of these numbers. So after evaluating, now we need to multiply and simplify, and we get 63. And 63 divided by negative three is negative 21.
So after solving this system using determinants, 𝑥 equals three, 𝑦 equals negative 25, and 𝑧 equals negative 21.