It is often desirable or even necessary to use more than one variable to model a situation in many fields. When this is the case, we write and solve a system of equations in order to answer questions about the situation.

If a system of linear equations has at least one solution, it is **consistent**. If the system has no solutions, it is **inconsistent**. If
the system has an infinity number of solutions, it is **dependent**.
Otherwise it is **independent**.

A linear equation in three variables describes a plane and is an equation
equivalent to the equation

where A, B, C, and D are real numbers and A, B, C, and D are not all 0.

**Example 2:**

The standard equation of a circle is
Find the
equation of the circle that passes through the points
,
, and

Let's create three equations from the given points.

(1) | |||

(2) | |||

(3) | |||

We are going to show you how to solve this system of equations three different ways:

1) Substitution,
2) Elimination
3) Matrices

**SUBSTITUTION**:

The process of substitution involves several steps:

Step 1: Solve for one of the variables in one of the equations. It
makes no difference which equation and which variable you choose. Let's
solve for in equation (1).

Step 2: Substitute this value for in equations (2) and (3). This will change equations (2) and (3) to equations in the two variables and . Call the changed equations (4) and (5), respectively.

(4) | |||

(5) | |||

Step 3: Solve for in equation (4).

Step 4: Substitute this value of in equation (5). This will give you an equation in one variable.

Step 5: Solve for .

Step 6: Substitute this value of in equation (4) and solve for

Step 7: Substitute for and for in equation (1) and solve for .

The solution: The equation of the circle that contains the points , , and is

Step 8: Check the solutions:

The process of elimination involves several steps: First you reduce three
equations to two equations with two variables, and then to one equation with
one variable.

Step 1: Decide which variable you will eliminate. It makes no
difference which one you choose. Let us eliminate first..

Step 2: Add equations (1) and (2) to form equation (4), then add equations (2) and (3) to form equation (5). Equations (4) and (5) will contain the variables A and B.

Step 3: We now have two equations with two variables. Let's simplify these two equations.

Step 4: Add the simplified equations (4) and (5) to create equation (6) with just one variable.

Step 6: Substitute for in equation (4) and solve for B.

Step 7: Substitute for and for in equation (1) and solve for .

The equation of the circle is Check your answers as before.

The process of using matrices is essentially a shortcut of the process of
elimination. Each row of the matrix represents an equation and each column
represents coefficients of one of the variables.

Step 1:
Create a three-row by four-column matrix using coefficients and the constant
of each equation.

The vertical lines in the matrix stands for the equal signs between both
sides of each equation. The first column contains the coefficients of ,
the second column contains the coefficients of , the third column
contains the coefficients of , and the last column contains the constants
to the right of the equal signs.

We want to convert the original matrix

to the equivalent matrix.

Then you can read the matrix as , and .

Step 2: We work with column 1 first. We want a 1 in Cell 11 [Row 1-Col
1]. To achieve this, multiply Row 1 by to form a new Row 1.

Step 3: Add -2 times Row 1 to Row 2 to form a new Row 2, and add -6 times Row 1 to Row 3 to form a new Row 3.

Step 4: Let's now manipulate the matrix so that there is a 1 in Cell 22. We do this by multiplying Row 2 by

Step 5: Let's now manipulate the matrix so that there are zeros in Cell 12 and Cell 32. We do this by adding times Row 2 to Row 1 to form a new Row 1 and by adding times Row 2 to Row 3 to form a new Row 3.

Step 6: Let's now manipulate the matrix so that there is a 1 in Cell 33. We do this by multiplying Row 3 by

Step 7: Let's now manipulate the matrix so that there are zeros in
Cells 13 and 12.

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**.

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**Author**: Nancy Marcus

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