Concepts in Advanced Algebra

Instruction 1-5

What are Matrices | Adding Matrices | Subtracting Matrices | Determinant | Solving Systems of Equations Using Matrices | Summary

We can solve systems of linear equations using determinants. Here, we learn how to solve the systems of two linear equations and systems of three linear equations. Such method is called Cramer’s Rule for solving systems of linear equations.

Solving a System of Two Linear Equations in Two Variables Using Cramer’s Rule

Let be the general form of a system of two linear equations in two variables. To solve such a system using Cramer’s rule, we must find the following determinants formed by the coefficients and constants in the system.

Practice 14. Solve the system of equations below using Cramer’s rule.

x + y = 5

2x – y = 1

Solution

Set determinants A, D_x, and D_y using the coefficients and constants.

Solving a System of Three Linear Equations in Three Variables Using Cramer’s Rule

Then, replace the known values in the following relationships to find x, y, and z.

Real Life Application . Collins Company manufactures sofas and armchairs. During year 2003, it manufactured a total of 480,000 sofas and armchairs. If the number of the armchairs produced were 11200 more than the number of sofas, find the number of each product manufactured during year 2003.

Solution. Denote the number of sofas by x and the number of armchairs by y. Then,

x + y = 480,000

x + 11200 = y

Write the system of equations in standard form as below.

x + y = 480,000

x – y = –11,200

Now, we can set the following determinants and find their values.

Practical Exercise 4. Solve each system of equations using Cramer’s Rule.

Answer

Video Instruction
*Availability of You Tube video links may vary. eTAP has no control of these materials.

• Solving a 3 X 3 System of Equations Using Cramer’s Rule [5:14]

• Solving a 3 X 3 System of Equations Using Cramer’s Rule [8:19]

• Solving a 2 X 2 System of Equations Using Cramer’s Rule [5:53]

for Students, Parents and Teachers

Matrices. A matrix (plural: matrices) is a rectangular array of numbers. Each of these numbers is called an entry or an element. The elements of a matrix usually are enclosed by two brackets.

The horizontal lines of a matrix are called rows and the vertical lines are called columns. The size of a matrix is identified by the number of its rows and columns. The size of a matrix denoted as an index for the label of the matrix. The index usually is in the form of m × n in which m is the number of rows and n is the number of columns.

Square Matrix. If a matrix has the same number of rows and columns, it is called a square matrix.

Adding Matrices. Adding two or more matrices is a simple task. They just must have the same size. Then, adding the corresponding entries of the given matrices results in the addition of the matrices.

Subtracting Matrices. Subtracting two matrices is similar to addition of two matrices. They must be of the same size. To find the subtraction of two matrices simplify subtract the corresponding entries and place the results as the entries of a new matrix. This matrix will be the result of subtracting two matrices.

Determinants. Only square matrices have determinants. Determinant of a matrix is shown by placing the entries of the matrix between two vertical bars |  |.

Determinant of a 2 × 2 Square Matrix. For a 2 × 2 square matrix

Solving Systems of Equations Using Matrices. We can solve systems of linear equations using determinants. Here, we learn how to solve the systems of two linear equations and systems of three linear equations. Such method is called Cramer’s Rule for solving systems of linear equations.

Solving a System of Two Linear Equations in Two Variables Using