SYSTEM SOLVERS

Solving $A\mathbf{{x}} = \mathbf{{b}}$ . Gaussian Elimination, Row Reduction, and the algorithm of absolute truth.

01 // The Setup

We don't solve systems of equations by juggling variables anymore. We detach the coefficients and put them into an Augmented Matrix, treating the entire system as a singular object.

EquationsMatrix Form

2x + 3y = 5

1x - 1y = 0

\left[ \begin{array}{cc|c} 2 & 3 & 5 \\ 1 & -1 & 0 \end{array} \right]

02 // The Toolset

Row Op 1

Swap

Exchange any two rows. Changing the order of information does not change the truth.

R_1 \leftrightarrow R_2

Row Op 2

Scale

Multiply a row by a non-zero number. Used to turn the leading pivot number into a 1.

R_1 \leftarrow \frac{1}{2} R_1

Row Op 3

Eliminate

Add a multiple of one row to another. Used to destroy variables (turning them into a 0).

R_2 \leftarrow R_2 - 3R_1

03 // The Execution

RREF Target

The algorithm is complete when you achieve the Identity Matrix on the left side of the augment. The numbers remaining on the right side are your final answers. Step through the lab below to watch Gaussian Elimination in real-time.

Gaussian Elimination Engine

Step 1 / 5

\left[ \begin{array}{cc|c} 2 & 3 & 5 \\ 1 & -1 & 0 \end{array} \right]

The Starting Matrix

We extract the coefficients into an augmented matrix.

SYSTEM SOLVED: x = 1, y = 1

Systems Solved

You are ready to command N-dimensional space.

Next: Determinants