### Question Description

I’m studying for my Calculus class and need an explanation.

- Write a problem for a classmate to solve that can be translated to a system of two (2) or more equations in at least two (2) variables. Explain your answer.

## Final Answer

Thank you for the opportunity to help you with your question!

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:**

Let's create three equations from the given points.

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 [img width="17" height="15" align="BOTTOM" border="0" src="http://www.sosmath.com/soe/SE3001/img21.gif" alt="$C$"> in equation (1).

Step 2: Substitute this value for [img width="17" height="15" align="BOTTOM" border="0" src="http://www.sosmath.com/soe/SE3001/img21.gif" alt="$C$" > 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 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:

**ELIMINATION**:

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.

**MATRICES**:

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

We want to convert the original matrix

to the equivalent matrix.

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

You can now read the answers off the matrix: , , and . check your answers by the method described above.