# Precalculus : Solve a System of Equations in Three Variables Using Augmented Matrices

## Example Questions

### Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Express this system of equations as an augmented matrix:

Explanation:

Arrange the equations into the form:

, where a,b,c,d are constants.

Then we have the system of equations: .

The augmented matrix is found by copying the constants into the respective rows and columns of a matrix.

The vertical line in the matrix is analogous to the = sign thus resulting in the following:

### Example Question #2 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Using an augmented matrix, solve the following system of equations:

Which of the following are the values of ,,and  that satisfy this system?

Explanation:

Create an augmented matrix by entering the coefficients into one matrix and appending a vector to that matrix with the constants that the equations are equal to. Then you can row reduce to solve the system.

First, lets make this augmented matrix:

Now we can row reduce the matrix using the three row reduction operations: mutliply a row, add one row to another, swap row positions.

First, we can subtract

Swap

Subtract

Subtract

Mulitply

You can stop here given that this augmented matrix can be rewritten as a system again with

, or you can continue using the matrix, subtracting multiples of  from the other two rows to get an identity matrix yielding the solution to the system.

### Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

True or false: there is no  solution  that makes this matrix equation true.

True

False

False

Explanation:

For two matrices to be equal, two conditions must hold:

1) The matrices must have the same dimension. This can be seen to be the case, since both matrices have two rows and one column.

2) All corresponding entries must be equal. For this to happen, it must hold that

This is a system of two equations in two variables, which can be solved as follows:

Add both sides of the equations:

It follows that

Substitute back:

Thus, there exists a solution to the system, and, consequently, to the original matrix equation. The statement is therefore false.

### Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

and .

True or false: , where is the two-by-two identity matrix.

False

True

True

Explanation:

First, it must be established that  is defined. This is the case if and only and have the same number of rows, which is true, and they have the same number of columns, which is also true.  is therefore defined.

This is the two-by-two identity . Therefore, .

### Example Question #5 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Using the Reduced Row Echelon Method, the solution matrix to the system of equations reads

Which of the following is the accurate depiction of this matrix?

Explanation:

Traditionally in Row Reduced Echelon Form (first row's values are 1,0,0; second row's values are 0,1,0; third row's values are 0,0,1), x is depicted in the first row,  in the second row, and  in the third. Therefore the correct answer is .

### Example Question #6 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Determine which  matrix represents the following system of equations:

Explanation:

The system of equations in matrix form is written:

Beginning with the top equation, insert each coefficient into the top row of the matrix, from left to right. Then, place the 2 on the right hand side of the vertical line (anytime you see this vertical line in a matrix, it is called an augmented matrix. Because the first equation's coefficients are 1, 1, and 1, these form the top row of the matrix, with 2 on the right side of the bar. Then, move on to the second row, which has coefficients 2, -1, and 3. Place these into the second row of the matrix, with -7 on the right hand side of the vertical bar. Finally, take the coefficients from the 3rd equation, 1, 1, and -1, and place these in the bottom row with 6 on the right hand side of the vertical bar.

### Example Question #7 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Determine which 3x3 matrix represents the following system of equations:

;

;

.