Exercise 32 Page 180

To solve a system of equations using a matrix on the calculator, begin by pushing the $\boxed{2\text{ND}}$ button, followed by $\boxed{x^{\text{-}1}}.$ Then, scroll to the right until you reach the EDIT menu. This will bring up a list of all available matrices. Please note that the dimensions shown on the right will depend on your previous use of matrices in your calculator.

For this exercise, we want a $3\times 4$ matrix. This is option $\left[\text{A}\right]$ in the list. Next, fill in the matrix with the coefficients and constants of each of the three equations. There are three things we need to remember when filling in the matrix.

1. If a variable is being subtracted, the coefficient is negative.
2. If a variable is missing, the coefficient is $0.$
3. If a variable does not have a written coefficient, the coefficient is $1.$

Let's see what this will look like for the given system.

Next, to solve the system of equations, press $\boxed{2\text{ND}}$ and $\boxed{x^{\text{-}1}}$ once more. However, this time we will then select the MATH menu and scroll down until we find the rref( option. Press $\boxed{\text{ENTER}}$ to select the function.

Finally, we assign matrix $\left[\text{A}\right]$ to the rref( function. Again we push $\boxed{2\text{ND}}$ and $\boxed{x^{\text{-}1}}$ to bring up all available matrices. In the NAMES menu, choose matrix $\left[\text{A}\right]$ by pushing $\boxed{\text{ENTER}}.$ If we push $\boxed{\text{ENTER}}$ one more time, the calculator will solve the system of equations.

The solutions can be read vertically in the last column, using the remaining coefficients as reference points for the variables. We can now see that the solution is $(1,1,0).$