Learn on PengienVision, Algebra 2Chapter 1: Linear Functions and Systems

Lesson 7: Solving Linear Systems Using Matrices

In this Grade 11 enVision Algebra 2 lesson, students learn how to solve systems of linear equations using augmented matrices and matrix row operations, including switching rows, multiplying or dividing rows by a nonzero constant, and adding or subtracting rows. The lesson guides students through transforming a matrix into reduced row echelon form to identify the unique solution of both two-variable and three-variable linear systems. This foundational skill connects algebraic equation-solving methods to structured matrix procedures used throughout higher-level mathematics.

Section 1

Row Operations

Property

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

Interchange any two rows. ( $R_i \leftrightarrow R_j$ )
Multiply a row by any real number except 0. ( $kR_i$ )
Add a nonzero multiple of one row to another row. ( $R_i + kR_j$ )

Examples

Given the matrix $\left[\begin{array}{cc|c} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right]$ :

Interchanging rows 1 and 2 ( $R_1 \leftrightarrow R_2$ ) results in $\left[\begin{array}{cc|c} 2 & 4 & 6 \\ 1 & 3 & 5 \end{array}\right]$ .

Multiplying row 2 by $3$ ( $3R_2$ ) results in $\left[\begin{array}{cc|c} 1 & 3 & 5 \\ 6 & 12 & 18 \end{array}\right]$ .

Section 2

Reduced Row-Echelon Form

Property

For a consistent and independent system of equations, its augmented matrix is in reduced row-echelon form when to the left of the vertical line, each entry on the diagonal is a $1$ , all entries below the diagonal are zeros, and all entries above each leading $1$ are also zeros.

\begin{bmatrix} 1 & 0 & 0 & | & a \\ 0 & 1 & 0 & | & b \\ 0 & 0 & 1 & | & c \end{bmatrix}

Examples

Section 3

Systematic Row Reduction Process

Property

To systematically achieve reduced row echelon form (RREF), work column by column from left to right: (1) Create a leading 1 in the pivot position, (2) Use the pivot row to eliminate all other entries in that column, (3) Move to the next column and repeat until the matrix is in the form where each leading 1 is the only nonzero entry in its column.

Examples

Section 4

Solve systems using matrices and reduced row-echelon form

Property

To solve a system of equations using matrices:

Write the augmented matrix for the system of equations.
Using row operations, get the entry in row 1, column 1 to be 1.
Using row operations, get zeros in column 1 above and below the 1.
Using row operations, get the entry in row 2, column 2 to be 1.
Using row operations, get zeros in column 2 above and below the 1.
Continue the process until the matrix is in reduced row-echelon form.
Read the solution directly from the final matrix.
Write the solution as an ordered pair or triple.
Check that the solution makes the original equations true.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Row Operations

Property

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

Interchange any two rows. ( $R_i \leftrightarrow R_j$ )
Multiply a row by any real number except 0. ( $kR_i$ )
Add a nonzero multiple of one row to another row. ( $R_i + kR_j$ )

Examples

Given the matrix $\left[\begin{array}{cc|c} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right]$ :

Interchanging rows 1 and 2 ( $R_1 \leftrightarrow R_2$ ) results in $\left[\begin{array}{cc|c} 2 & 4 & 6 \\ 1 & 3 & 5 \end{array}\right]$ .

Multiplying row 2 by $3$ ( $3R_2$ ) results in $\left[\begin{array}{cc|c} 1 & 3 & 5 \\ 6 & 12 & 18 \end{array}\right]$ .

Section 2

Reduced Row-Echelon Form

Property

\begin{bmatrix} 1 & 0 & 0 & | & a \\ 0 & 1 & 0 & | & b \\ 0 & 0 & 1 & | & c \end{bmatrix}

Examples

Section 3

Systematic Row Reduction Process

Property

Examples

Section 4

Solve systems using matrices and reduced row-echelon form

Property

To solve a system of equations using matrices:

Write the augmented matrix for the system of equations.
Using row operations, get the entry in row 1, column 1 to be 1.
Using row operations, get zeros in column 1 above and below the 1.
Using row operations, get the entry in row 2, column 2 to be 1.
Using row operations, get zeros in column 2 above and below the 1.
Continue the process until the matrix is in reduced row-echelon form.
Read the solution directly from the final matrix.
Write the solution as an ordered pair or triple.
Check that the solution makes the original equations true.

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Row Operations

Property

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

Interchange any two rows. ( $R_i \leftrightarrow R_j$ )
Multiply a row by any real number except 0. ( $kR_i$ )
Add a nonzero multiple of one row to another row. ( $R_i + kR_j$ )

Examples

Given the matrix $\left[\begin{array}{cc|c} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right]$ :

Interchanging rows 1 and 2 ( $R_1 \leftrightarrow R_2$ ) results in $\left[\begin{array}{cc|c} 2 & 4 & 6 \\ 1 & 3 & 5 \end{array}\right]$ .

Multiplying row 2 by $3$ ( $3R_2$ ) results in $\left[\begin{array}{cc|c} 1 & 3 & 5 \\ 6 & 12 & 18 \end{array}\right]$ .

Section 2

Reduced Row-Echelon Form

Property

\begin{bmatrix} 1 & 0 & 0 & | & a \\ 0 & 1 & 0 & | & b \\ 0 & 0 & 1 & | & c \end{bmatrix}

Examples

Section 3

Systematic Row Reduction Process

Property

Examples

Section 4

Solve systems using matrices and reduced row-echelon form

Property

To solve a system of equations using matrices:

Write the augmented matrix for the system of equations.
Using row operations, get the entry in row 1, column 1 to be 1.
Using row operations, get zeros in column 1 above and below the 1.
Using row operations, get the entry in row 2, column 2 to be 1.
Using row operations, get zeros in column 2 above and below the 1.
Continue the process until the matrix is in reduced row-echelon form.
Read the solution directly from the final matrix.
Write the solution as an ordered pair or triple.
Check that the solution makes the original equations true.