How to Perform Gaussian Elimination: Step-by-Step Instructions

Gaussian elimination is a method for solving systems of linear equations using row operations to achieve row echelon form. While our Gaussian Elimination Calculator does this for you quickly, understanding the manual process helps you see the logic behind it. This step-by-step guide walks you through the process by hand, so you can solve systems without a computer. For a broader overview, check out What Is Gaussian Elimination? Definition, Steps & Examples 2026.

You'll Need

• Paper and pencil (or pen)
• Eraser (for corrections)
• Basic arithmetic skills (addition, subtraction, multiplication, division)
• Understanding of matrix notation: an augmented matrix [A|b] where A is the coefficient matrix and b is the constants column
• Optional: a scientific calculator to double-check your fraction arithmetic

Step-by-Step Process

1. Write the augmented matrix. Given a system of linear equations, write the coefficients and constants in matrix form: each row represents an equation, columns are variables (usually x, y, z, ...), and the last column after the vertical bar is the constants.
2. Perform forward elimination to reach row echelon form (REF). Use elementary row operations to create zeros below the first pivot (top-left element). The three allowed operations are: row switching, row multiplication by a non-zero constant, and row addition (adding a multiple of one row to another). Aim for a staircase pattern of pivots (leading 1s or non-zero numbers).
3. Continue elimination for remaining columns. Move to the second column, second row. Create a pivot there (if possible) and eliminate below it. Repeat for all columns until you have an upper triangular matrix (or until you run out of rows/columns).
4. Back substitution (if REF) or further reduction to RREF. Once in REF, solve from the last equation upward: substitute known values into the equation above. Alternatively, continue with row operations to get reduced row echelon form (RREF) where each pivot is 1 and all entries above pivots are zero—then the solution is read directly. For manual calculation, back substitution is often easier.
5. Interpret the result. The solution may be a unique set of values, infinitely many (parameterized), or none (inconsistent). For more on this, see What Do Gaussian Elimination Results Mean? Solution Types.

Example 1: 2x2 System

Solve:
2x + 3y = 5
4x - y = 1

Step 1: Augmented matrix

[ 2   3  |  5 ]
[ 4  -1  |  1 ]

Step 2: Eliminate below first pivot (2). We want to zero out the (2,1) entry. Replace Row2 with Row2 - 2*Row1 (since 4/2 = 2).
Row2 new: (4-2*2) = 0; (-1 - 2*3) = -7; (1 - 2*5) = -9.
New matrix:

[ 2   3  |  5 ]
[ 0  -7  | -9 ]

Step 3: Back substitution. From Row2: -7y = -9 → y = 9/7.
Substitute into Row1: 2x + 3*(9/7) = 5 → 2x + 27/7 = 5 → 2x = 5 - 27/7 = 35/7 - 27/7 = 8/7 → x = 4/7.
Solution: x = 4/7, y = 9/7.

Example 2: 3x3 System

Solve:
x + y + z = 6
2x - y + z = 3
3x + 2y - z = 4

Step 1: Augmented matrix

[ 1   1   1  |  6 ]
[ 2  -1   1  |  3 ]
[ 3   2  -1  |  4 ]

Step 2: Eliminate below first pivot (1). Row2 = Row2 - 2*Row1; Row3 = Row3 - 3*Row1.
Row2 new: (2-2)=0; (-1-2)= -3; (1-2)= -1; (3-12)= -3.
Row3 new: (3-3)=0; (2-3)= -1; (-1-3)= -4; (4-18)= -14.
Matrix:

[ 1   1   1  |  6 ]
[ 0  -3  -1  | -3 ]
[ 0  -1  -4  | -14 ]

Step 3: Eliminate below second pivot (-3). Use Row2 as pivot row. To zero out Row3's second column: Row3 = Row3 - (1/3)*Row2? Actually, easier: Row3 = 3*Row3 - Row2 (to avoid fractions temporarily). But careful: that multiplies Row3 by 3, which changes determinant but not solution. Alternatively, use Row2 to eliminate directly. Let's do Row3 = Row3 - (1/3)*Row2 to keep numbers smaller. But fractions appear. Instead, we'll use integer operations: Row3 = 3*Row3 - Row2.
New Row3: (3*0 - 0)=0; (3*(-1) - (-3)) = -3+3=0; (3*(-4) - (-1)) = -12+1= -11; (3*(-14) - (-3)) = -42+3= -39.
Matrix:

[ 1   1   1  |  6 ]
[ 0  -3  -1  | -3 ]
[ 0   0  -11 | -39 ]

Step 4: Back substitution. From Row3: -11z = -39 → z = 39/11 = 3.545... but as fraction z=39/11 simplifies? 39/11 = 3.545... keep fraction.
Row2: -3y - z = -3 → -3y - 39/11 = -3 → -3y = -3 + 39/11 = -33/11 + 39/11 = 6/11 → y = -2/11 (since dividing by -3: 6/11 / -3 = -2/11).
Row1: x + y + z = 6 → x - 2/11 + 39/11 = 6 → x + 37/11 = 6 → x = 6 - 37/11 = 66/11 - 37/11 = 29/11.
Solution: x = 29/11, y = -2/11, z = 39/11.

Common Pitfalls

• Dividing by zero: Never divide a row by zero. If a pivot is zero, try swapping rows.
• Forgetting to apply the operation to the entire row: Including the constant column—a common mistake.
• Sign errors: When subtracting a multiple of a row, double-check signs.
• Mishandling fractions: Keep fractions in simplest form; avoid rounding until the final step.
• Inconsistent systems: If you get a row like [0 0 0 | 5], the system has no solution. If you get a row of zeros with zero constant, there may be infinite solutions. Visit our solution types page for more details.

For more on the underlying matrix operations, see Gaussian Elimination Formula: Matrix Row Operations Explained. After mastering manual calculations, use the Gaussian Elimination Calculator to verify your work and save time on larger systems.