Use matrices to solve the system.
This pair of equations can be expressed as a matrix equation. So what we need to do is we need to take the coefficients and put them into a matrix and then multiply by the 𝑥𝑦-matrix and then set it equal to the answers, eight and eight. So it’s set equal to the solutions, eight and eight, in a matrix.
So to solve this matrix equation, multiply both sides by the multiplicative inverse of the matrix of coefficients. So what does that mean? We are trying to get 𝑥 and 𝑦 alone cause we wanna solve for 𝑥 and 𝑦. So how do we get rid of that matrix that’s made up of the coefficients. Well, we need the inverse of that matrix, so if we multiply the left side by the inverse matrix, it will make that coefficient matrix wipe out and be gone. But, whatever we do to one side of the equation, we have to do to the other. So we would have to multiply it to the right-hand side as well.
So in order to find the inverse of this matrix, we need to use a formula. And the formula for the general matrix 𝑎𝑏𝑐𝑑, to find the inverse of that, we take one divided by 𝑎𝑑 minus 𝑏𝑐 times the matrix 𝑑, negative 𝑏, negative 𝑐, 𝑎. So 𝑎 is equal to negative one. So we can replace all of the the 𝑎s with negative one. Next, 𝑏 is equal to five. Now that we’ll replace 𝑏 with five, 𝑐 is equal to negative three. And then lastly, 𝑑 is equal to one.
So now we need to evaluate and simplify. So we have one divided by negative one minus negative 15, so truly negative one plus 15. And then in our matrix, negative negative three reduce to positive three. So, so our inverse matrix is equal to one 14th times one, negative five, three, negative one.
So now we need to go ahead and plug it in. So we know the inverse matrix will cancel with the normal matrix on the left, so we don’t really need to plug that in. But on the right-hand side, we do. And it’s okay to go ahead and write the one 14th over to the right. So that just means whatever we get, we’ll multiply every single number in that matrix by one 14th.
So after simplifying, now I need to multiply these matrices. So we take eight times one plus eight times negative five, and then eight times three plus eight times negative one. And then as stated before, each of these numbers, whatever we get, we will multiply by one 14th. So we have eight minus 40 and then 24 minus eight. So we need to take negative 32 times one 14th and 16 times one 14th. Now after reducing, we get negative 16 17ths and eight-sevenths.
And remember, this is equal to 𝑥, 𝑦. So 𝑥 is equal to negative 16 17ths and 𝑦 is equal to eight-sevenths.