Consider the following simultaneous equations: three 𝑥 plus two 𝑦 is equal to eight; negative eight 𝑥 minus nine 𝑦 is equal to two. Write the determinant Δ sub 𝑥. Write the determinant Δ sub 𝑦. Write the determinant Δ.
In this question, we’re given a pair of simultaneous equations in two variables. And we’re asked to express three different determinants: Δ sub 𝑥, Δ sub 𝑦, and Δ. And we can recall that these three matrices are involved in Cramer’s rule. So let’s start by recalling how we find the determinant Δ sub 𝑥. We want to form a square matrix involving the coefficients of 𝑥, 𝑦 and the constant values. Of course, we can’t do this directly since there are six values. Instead, we need to note that we have Δ sub 𝑥. And we can recall the sub 𝑥 in this expression tells us that we need to substitute the column for the constants in this equation into the column for the 𝑥-coefficients in our coefficient matrix.
So our first column will be the column for the constants; that’s the column eight, two. Then the second column of this matrix will be the coefficients of 𝑦, where we need to remember to include the negative sign. That’s the column two, negative nine, which is then our final answer. Δ sub 𝑥 is the determinant of the two-by-two matrix eight, two, two, negative nine, which is also the determinant of the coefficient matrix, where we substitute the column for the constants of this equation for the column for the coefficients of 𝑥.
We can follow the exact same process to find the determinant Δ sub 𝑦. The first column in this matrix will be the coefficients of 𝑥. That’s three, negative eight. Next, since we’re finding Δ sub 𝑦, our next column is not the coefficients of 𝑦. Instead, it’s the constants for the equation. That’s eight, two, which is then our final answer. Δ sub 𝑦 is equal to the determinant of the two-by-two matrix three, eight, negative eight, two.
Finally, we need to find the determinant Δ. Remember, Δ is the determinant of the matrix of coefficients. That’s just the coefficients of 𝑥 and 𝑦. So the first column will be the coefficients of 𝑥; that’s three, negative eight. And the second column is the coefficients of 𝑦; that’s two, negative nine, which then gives us our final answer. Δ is equal to the determinant of the two-by-two matrix three, two, negative eight, negative nine.
But before we end, it’s worth noting we can use these to solve our simultaneous equations. Cramer’s rule tells us if the determinant of the matrix of coefficients Δ is nonzero, then 𝑥 is equal to Δ sub 𝑥 divided by Δ and 𝑦 is equal to Δ sub 𝑦 divided by Δ is the unique solution to the system. In other words, if we evaluate the determinants of these three matrices, we can use this to find the unique solution to this system of simultaneous equations. However, this was not necessary to answer our question, although it could be a useful check to make sure that our expressions are correct.