Question Video: Using the Determinants in Cramer's Rule to Find the System of Equations Mathematics

Video Transcript

According to Cramer’s rule and given that △ sub 𝑥 is equal to the determinant of the two-by-two matrix three, seven, five, two and △ sub 𝑦 is equal to the determinant of the two-by-two matrix negative five, three, negative two, five, write the simultaneous equations of the system.

In this question, we’re given expressions for △ sub 𝑥 and △ sub 𝑦 in accordance with Cramer’s rule. We need to use these expressions to determine the simultaneous equations of the system represented by these two expressions. And to do this, we start by noting we’re given expressions for △ sub 𝑥 and △ sub 𝑦. And these are given as two-by-two matrices. This means our system will only have two unknowns, 𝑥 and 𝑦.

So let’s recall Cramer’s rule for two unknowns. This tells us if we have two simultaneous equations 𝑎𝑥 plus 𝑏𝑦 is equal to 𝑒 and 𝑐𝑥 plus 𝑑𝑦 is equal to 𝑓 and the determinant of the matrix of coefficients is nonzero — that’s △ is equal to the determinant of the matrix 𝑎, 𝑏, 𝑐, 𝑑 — then 𝑥 is equal to △ sub 𝑥 divided by △ and 𝑦 is equal to △ sub 𝑦 divided by △ is the unique solution to the system of linear equations.

Well, it’s worth noting △ sub 𝑥 is the determinant of the matrix 𝑒, 𝑏, 𝑓, 𝑑 and △𝑦 is the determinant of the matrix 𝑎, 𝑒, 𝑐, 𝑓. So to find an expression for △ sub 𝑥, we take our expression for △ and replace the first column in the expression for △, which is the coefficients of 𝑥 in our simultaneous equations, with the constants of the simultaneous equations. That’s 𝑒, 𝑓.

So to find the simultaneous equations, we need to determine the values of 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, and 𝑓. And we can do this by using the given expressions for △ sub 𝑥 and △ sub 𝑦. First, using △ sub 𝑥, we can determine the values of 𝑒, 𝑏, 𝑓, and 𝑑. We have 𝑒 is equal to three, 𝑏 is equal to seven, 𝑓 is equal to five, and 𝑑 is equal to two. And we can then follow the same process for our expression for △ sub 𝑦. This gives us that 𝑎 is equal to negative five, 𝑒 is equal to three, 𝑐 is equal to negative two, and 𝑓 is equal to five.

And now we can just substitute these values into our simultaneous equations, which then gives us our final answer. The system of simultaneous equations given by Cramer’s rule for △ sub 𝑥 and △ sub 𝑦 given in the question is negative five 𝑥 plus seven 𝑦 is equal to three and negative two 𝑥 plus two 𝑦 is equal to five.