A system of equations is a collection of two or more equations with the same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.

The equations in the system can be linear or non-linear. This tutorial
reviews systems of linear equations.

A problem can be expressed in narrative form or the problem can be expressed
in algebraic form.

Let's start with an example stated in narrative form. We'll convert it to an
equivalent equation in algebraic form, and then we will solve it.

**Example 4:**

The perimeter of a rectangle is 34 inches. If the length is 2 inches more
than twice the width, find the length and the width of the rectangle.

The perimeter of a rectangle is the sum of the four sides.

**Solution:
**

**
**
We have two unknowns: the length and the width.

Sentence (1):''The perimeter of a rectangle is 34 inches...'' can be
restated as width + width + length + length =34.

Sentence (2): ''... the length is 2 inches more than twice the width ... can
be restated as the length is two times the width plus 2.

It is going to get tiresome writing the two words (length) and (width) over
and over again. So let's write them in shortcut form. Call the word (length)
by the symbol and call the word (width) by the symbol .

Let's rewrite sentences (1) and (2) in shortcut form.

We have converted a narrative statement of the problem to an equivalent
algebraic statement of the problem. Let's solve this system of
equations.

A system of linear equations can be solved four different ways:

Substitution,

Elimination,

Matrices,

Graphing.

**The Method of Substitution:**

The method of substitution involves several steps:

Step 1:

Simplify equation (1) and solve for in equation (1).

Step 2:

Substitute this value for in equation (2). This will change equation (2)
to an equation with just one variable, .

Step 3:

Solve for in the translated equation (2).

Step 4:

Substitute this value of in equation (1).

Step 5:

Check your answers by substituting the values of and in each of the
original equations. If, after the substitution, the left side of the
equation equals the right side of the equation, you know that your answers
are correct.

The process of substitution involves several steps:

In a two-variable problem rewrite the equations so that when the equations
are added, one of the variables is eliminated, and then solve for the
remaining variable.

Simplify both equations.

Step 1:

Change equation (2) by multiplying equation (2) by to obtain a new and equivalent
equation (2).

Step 2:

Add new equation (2) to equation (1) to obtain equation (3).

Step 3:

Substitute in equation (1) and solve for.

Step 4:

Check your answers (See above check).

**The Method of Matrices:**

This method is essentially a shortcut for the method of elimination.

Rewrite equations (1) and (2) without the variables and operators. The left
column contains the coefficients of the L's, the middle column contains the
coefficients of the W's, and the right column contains the
constants.

The objective is to reorganize the original matrix into one that looks like

Step 1.

Manipulate the matrix so that the number in cell 11 (row 1-col 1) is 1. In
this case, Multiply row 1 by 1/2.

Step 2:

Manipulate the matrix so that the number in cell 21 is 0. To do this we
rewrite the matrix by keeping row 1 and creating a new row 2 by adding -1 row 1 to row 2.

Manipulate the matrix so that the cell 22 is 1. Do this by multiplying row 2 by
1/3.

Manipulate the matrix so that cell 12 is 0. Do this by adding

**The method of Graphing:**

In this method solve for in each equation and graph both. The point of
intersection is the solution.

If you would like to work a similar example, click on
**Example**.

If you would like to test yourself by working some problem similar to this example, click on
**Problem**.

**
This site was built to accommodate the needs of students. The topics and problems are what
students ask for. We ask students to help in the editing so that future viewers will access
a cleaner site. If you feel that some of the material in this section is ambiguous or needs
more clarification, or if you find a mistake, please let us know by e-mail at sosmath.com.**

*
*

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

**Author**: Nancy Marcus

Contact us

Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA

users online during the last hour