Practise Gaussian elimination: solving linear systems

In Gaussian elimination you bring the augmented matrix to row echelon form with row operations and then substitute back from the bottom up. Here you do the work: you enter each row operation and the matrix changes at once.

How to do it

1
Write the system as an augmented matrix (A|b): one row per equation, the right-hand side after the bar.
2
Pick a non-zero pivot in the first column, ideally a 1 (swap rows if needed).
3
Create zeros below it: subtract a suitable multiple of the pivot row from each row, e.g. II − 3·I.
4
Repeat for the next columns until the matrix is in row echelon form.
5
Substitute back from the bottom up, or also eliminate above the pivots (Gauss–Jordan) until you can read off the solution.

Example with worked solution

Worked solution

Starting matrix
−2 3 3
−2 1 1
1. II − I
−2 3 3
0 −2 −2
2. −1/2·II
−2 3 3
0 1 1
3. I − 3·II
−2 0 0
0 1 1
4. −1/2·I
1 0 0
0 1 1

x₁ = 0, x₂ = 1

Your turn

Enter the row operations, e.g. II − 3·I. If you get stuck, "Hint: next step" helps.

Task 1 of 2Linear system · 2×2

Solve the system of linear equations.

Use row operations until you can read off the solution, then enter it below.

−2	1	−10
−4	0	−20

x₁, x₂ | b

Your answer

Typical mistakes

Forgetting the right-hand side: every row operation applies to the whole row, including the part after the bar.
Sign errors with negative numbers: if row I has a −2, II − 3·I gives +6 there.
Working with a row that has already changed without noticing: one operation at a time, always on the current matrix.
Multiplying a row by 0: that is not an allowed operation, an equation gets lost.

Frequently asked questions

Which row operations are allowed?

Swapping rows, multiplying a row by a non-zero number and adding a multiple of one row to another. These three operations do not change the set of solutions.

What is the difference between Gauss and Gauss–Jordan?

Gaussian elimination stops at row echelon form and then substitutes back. Gauss–Jordan also eliminates above the pivots and scales every pivot to 1, so the solution appears right after the bar.

How do I avoid fractions?

Use a 1 as pivot where possible (swap rows) or combine rows like 2·III − 3·I. If a row has a common factor, divide it out, e.g. ½·II.