Solve any linear system with rref( in under a minute

Systems of linear equations show up everywhere from algebra to physics to econ. Hand-elimination works for 2×2 but gets painful fast. The calculator's rref( (reduced row echelon form) function does the whole thing in one step.

Step 1: Enter the augmented matrix

Press 2nd + x⁻¹ (this is the MATRIX menu — the label above x⁻¹).

1. Arrow right to EDIT.
2. Pick any matrix name ([A] is fine).
3. Enter the dimensions. For a system of n equations in m unknowns, the augmented matrix is n × (m+1).

Example — solve this system:

2x + y − z = 8
-3x − y + 2z = -11
-2x + y + 2z = -3

This is a 3×4 matrix:

[ 2   1  -1 |  8 ]
[-3  -1   2 |-11 ]
[-2   1   2 | -3 ]

Enter dimensions 3 4, then type the 12 values row by row. Press 2nd + mode to quit out when done.

Step 2: Run rref(

On the home screen:

1. 2nd + x⁻¹ → arrow right to MATH → scroll to B:rref( (letter may vary by firmware).
2. Press enter. The function rref( appears on the home screen.
3. Tell it which matrix: 2nd + x⁻¹ → NAMES → 1:[A]. Close the paren: ). Press enter.

Reading the output

The calculator returns the matrix in reduced row echelon form:

[ 1  0  0 |  2 ]
[ 0  1  0 |  3 ]
[ 0  0  1 | -1 ]

Last column is your solution: x=2, y=3, z=-1.

Interpreting special outputs

• Last row is all zeros including the augmented column (e.g. [0 0 0 | 0]): infinite solutions. The system is under-determined; express one variable in terms of the others.
• Last row is [0 0 0 | c] with c ≠ 0: no solution. The system is inconsistent.
• Non-integer entries that look weird: apply ►Frac to see exact rational answers. rref([A])►Frac.

Why this beats substitution

For 3×3 and larger, Gauss-Jordan elimination is algorithmic — no insight required, no chance of an arithmetic slip halfway through. The calculator does the book-keeping perfectly.

Bonus: ref( vs rref(

• ref( gives row echelon form (upper-triangular, leading 1's). You still have to back-substitute.
• rref( goes all the way to reduced row echelon form — leading 1's with zeros above and below. The answer falls out.

Unless a teacher specifically asks for ref, use rref.

Gotcha

If your matrix has mixed fractions and decimals, the calculator may return floating-point results that look non-exact but are. Append ►Frac to force the exact fraction representation whenever you enter a system with fractional coefficients.