Systems of Linear Equations
Solve Systems of Equations Using Matrices
Learning Objectives
By the end of this section, you will be able to:
- Write the augmented matrix for a system of equations
- Use row operations on a matrix
- Solve systems of equations using matrices
Before you get started, take this readiness quiz.
Write the Augmented Matrix for a System of Equations
Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a matrix. A matrix is a rectangular array of numbers arranged in rows and columns.
A matrix is a rectangular array of numbers arranged in rows and columns.
A matrix with m rows and n columns has order The matrix on the left below has 2 rows and 3 columns and so it has order We say it is a 2 by 3 matrix.
Each number in the matrix is called an element or entry in the matrix.
We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.
Notice the first column is made up of all the coefficients of x, the second column is the all the coefficients of y, and the third column is all the constants.
Write each system of linear equations as an augmented matrix:
ⓐⓑ
ⓐ The second equation is not in standard form. We rewrite the second equation in standard form.
We replace the second equation with its standard form. In the augmented matrix, the first equation gives us the first row and the second equation gives us the second row. The vertical line replaces the equal signs.
ⓑ All three equations are in standard form. In the augmented matrix the first equation gives us the first row, the second equation gives us the second row, and the third equation gives us the third row. The vertical line replaces the equal signs.
Write each system of linear equations as an augmented matrix:
ⓐⓑ
ⓐ
ⓑ
Write each system of linear equations as an augmented matrix:
ⓐⓑ
ⓐ
ⓑ
It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.
Write the system of equations that corresponds to the augmented matrix:
We remember that each row corresponds to an equation and that each entry is a coefficient of a variable or the constant. The vertical line replaces the equal sign. Since this matrix is a , we know it will translate into a system of three equations with three variables.
Write the system of equations that corresponds to the augmented matri