# Solve a System of Equations on the TI-83 Plus

You can use the TI-83 Plus graphing calculator to solve a system of equations. Three matrices are associated with a system of linear equations: the coefficient matrix, the solution matrix, and the augmented matrix.

For example, **A**, **B**, and **C**, are (respectively) the coefficient matrix, solution matrix, and augmented matrix for this system of equations:

Systems of linear equations can be solved by first putting the augmented matrix for the system in reduced row-echelon form. The mathematical definition of reduced row-echelon form isn’t important here. It is simply an equivalent form of the original system of equations, which, when converted back to a system of equations, gives you the solutions (if any) to the original system of equations.

For example, when the reduced row-echelon matrix is converted to a system of equations, it gives the solutions *x* = -3, *y* = 3, and *z* = 9. The matrix converts to the system *x* – *z* = 0 and *y* – *z* = -2. This arrangement indicates that the system has an infinite number of solutions — namely, all solutions in which *x* = *z* and *y* = *z* – 2, where *z* is any real number.

The third picture illustrates a system that has no solution — the last line of the matrix says that 0 = 1, which is clearly impossible!

To solve a system of equations, follow these steps:

Define the augmented matrix in the Matrix editor.

You can define the coefficient and solution matrices for the system of equations and then augment these matrices to form the augmented matrix.

Press [2nd][MODE] to access the Home screen.

Press

You can also select the

**rref**command by pressingrepeatedly pressing

until the cursor is next to the

**rref**command, and then pressing**[ENTER]**.Enter the name of the matrix and then press [ ) ].

To enter the name of the matrix, press

**[2nd][**and key in the number of the matrix name. (On the TI-83, press*x*^{–1}]**[MATRX]**.)Press [ENTER] to put the augmented matrix in reduced row-echelon form.

To find the solutions (if any) to the original system of equations, convert the reduced row-echelon matrix to a system of equations.