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Solving a system of equations using a matrix (by hand)

Solving a system of equations using a matrix (by hand)

Author: Christopher Danielson

To demonstrate the computations involved in getting a 2 by 3 matrix into reduced row-echelon form.

This packet gives a quick overview of the process of solving systems of equations with matrices by hand.

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There are two main ways to solve systems of equations: by substitution and by elimination. Either one will work in all cases, but sometimes one technique is easier than the other.

But solving systems of equations by elimination is the process behind using matrices. A matrix is a set of rows and columns of numbers. Each number represents either a coefficient of a variable or a constant. In order to write a system of equations as a matrix, each equation first needs to be in standard form, as below:

Standard form for equations in two variables


Standard form for equations in three variables


The main reason for using a matrix to represent a system of equations is that a calculator or computer can then do the gruesome computations involved in solving the system.

In College Algebra classes, we commonly solve systems by hand and then use the calculator to solve more complicated systems using matrices. To save time in class, we sometimes skip all of the ugly details of the matrix computations that are being performed by the calculator.

This packet demonstrates those details. Enjoy.

Setting up the matrix

This video demonstrates the process of setting up a matrix from a system of equations.

Putting the matrix in reduced row-echelon form by hand

This video demonstrates the computations necessary to transform the original matrix into reduced row-echelon form. This form is the easiest one from which to interpret the solution to a system of equations.


So here is the good news...

Technology can run these computations for you. See my packets for solving systems of equations using matrices on a calculator for specific instructions on doing this with the TI-83, TI-84 or TI-89.