Chapter 7 - Systems of Linear Equations and Inequalities

Section 7.1 - Systems of Linear Equations

A system of linear equations (or simultaneous linear equations) is two or more linear equations.
A solution to a system of equations is the ordered pair or pairs that satisfy all equations in the system.
A system of linear equations may have exactly one solution, no solution, or infinitely many solutions.

Example - Is the Ordered Pair a Solution?

Determine which of the ordered pairs is a solution to the following system of linear equations.

3x – y = 2

4x + y = 5

a. (2, 4) b. (1, 1)

Procedure for Solving a System of Equations by Graphing

Determine three ordered pairs that satisfy each equation.
Plot the ordered pairs and sketch the graphs of both equations on the same axes.
The coordinates of the point or points of intersection of the graphs are the solution or solutions to the system of equations.

Graphing Linear Equations

When graphing linear equations, three outcomes are possible:

1.The two lines may intersect at one point, producing a system with one solution.

A system that has one solution is called a consistent system of equations.

2.The two lines may be parallel, producing a system with no solutions.

A system with no solutions is called an inconsistent system.

3.The two equations may represent the same line, producing a system with an infinite number of solutions.

A system with an infinite number of solutions is called a dependent system.

Example - Find the solution to the following system of equations graphically.

Section 7.2 - Solving Systems of Equations by the Substitution and Addition Methods

Procedure for Solving a System of Equations Using the Substitution Method

Solve one of the equations for one of the variables. If possible, solve for a variable with a coefficient of 1.
Substitute the expression found in step 1 into the other equation.
Solve the equation found in step 2 for the variable.
Substitute the value found in step 3 into the equation, rewritten in step 1, and solve for the remaining variable.

Example - Solve the following system of equations by substitution.

Addition Method

If neither of the equations in a system of linear equations has a variable with the coefficient of 1, it is generally easier to solve the system by using the addition (or elimination) method.
To use this method, it is necessary to obtain two equations whose sum will be a single equation containing only one variable.

Procedure for Solving a System of Equations by the Addition Method

1.If necessary, rewrite the equations so that the variables appear on one side of the equal sign and the constant appears on the other side of the equal sign.

2.If necessary, multiply one or both equations by a constant(s) so that when you add the equations, the result will be an equation containing only one variable.

3.Add the equations to obtain a single equation in one variable.

4.Solve the equation in step 3 for the variable.

5.Substitute the value found in step 4 into either of the original equations and solve for the other variable.

Example - Solve the system using the elimination method.

Section 7.3 –Matrices

A matrix is a rectangular array of elements.

An array is a systematic arrangement of numbers or symbols in rows and columns.

Matrices (the plural of matrix) may be used to display information and to solve systems of linear equations.
The numbers in the rows and columns of a matrix are called the elements of the matrix.
Matrices are rectangular arrays of numbers that can aid us by eliminating the need to write the variables at each step of the reduction.

For example, the system

may be represented by the augmented matrix

Dimensions of a Matrix

The dimensions of a matrix may be indicated with the notation rs, where r is the number of rows and s is the number of columns of a matrix.
A matrix that contains the same number of rows and columns is called a square matrix.

Example: 3  3 square matrices:

Example -

Addition and Subtraction of Matrices

Two matrices can only be added or subtracted if they have the same dimensions.
The corresponding elements of the two matrices are either added or subtracted.

Example - Find A + B

Multiplication of Matrices

A matrix may be multiplied by a real number, a scalar, by multiplying each entry in the matrix by the real number.
Multiplication of matrices is possible only when the number of columns in the first matrix is the same as the number of rows of the second matrix.
In general,

Example –

Identity Matrix in Multiplication

Example - Use the multiplicative identity matrix for a 2  2 matrix and matrix A to show that

The 2x2 identity matrix is

Multiplicative Identity Matrix

Square matrices have a multiplicative identity matrix.
The following are the multiplicative identity for a 2 by 2 and a 3 by 3 matrix.

For any square matrix A, AI = IA = A.

Section 7.4 - Solving Systems of Equations by Using Matrices

Augmented Matrix

The first step in solving a system of equations using matrices is to represent the system of equations with an augmented matrix.

An augmented matrix consists of two smaller matrices, one for the coefficients of the variables and one for the constant

Systems of equations Augmented Matrix

a1x + b1y = c1

a2x + b2y = c2

Row Transformations

To solve a system of equations by using matrices, we use row transformations to obtain new matrices that have the same solution as the original system.
We use row transformations to obtain an augmented matrix whose numbers to the left of the vertical bar are the same as the multiplicative identity matrix.

Procedures for Row Transformations

Any two rows of a matrix may be interchanged.
All the numbers in any row may be multiplied by any nonzero real number.
All the numbers in any row may be multiplied by any nonzero real number, and these products may be added to the corresponding numbers in any other row of numbers.

To Change an Augmented Matrix tothe Form

Use row transformations to:

1.Change the element in the first column, first row to a 1.

2.Change the element in the first column, second row to a 0.

3.Change the element in the second column, second row to a 1.

4. Change the element in the second column, first row to a 0.

Inconsistent and Dependent Systems

An inconsistent system occurs when, after obtaining an augmented matrix, one row of numbers on the left side of the vertical line are all zeros but a zero does not appear in the same row on the right side of the vertical line.
This indicates that the system has no solution.
If a matrix is obtained and a 0 appears across an entire row, the system of equations is dependent.

Triangularization Method

The triangularization method can be used to solve a system of two equations.
The ones and the zeros form a triangle.

In the previous problem we obtained the matrix

The matrix represents thefollowing equations.

x + 2y = 16

y = 7

Substituting 7 for y in the equation, then solving for x, x = 2.

Example -

Gauss-Jordan Elimination

Example - Solve the following system of equations by using matrices.

x + 2y = 16

2x + y = 11

Graphing Calculator-Matrices, Reduced Row Echelon Form

Example -

A System of Equations with an Infinite Number of Solutions

Example - Solve the system of equations given by

A System of Equations That Has No Solution

Example -Solve the system of equations given by

Systems with no Solution

If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a nonzero entry to the right of the line, then the system of equations has no solution.

The Acrosonic Company manufactures four different loudspeaker systems at three separate locations.

The company’s May output is as follows:

Model A / Model B / Model C / Model D
Location I / 320 / 280 / 460 / 280
Location II / 480 / 360 / 580 / 0
Location III / 540 / 420 / 200 / 880

If we agree to preserve the relative location of each entry in the table, we can summarize the set of data as follows:

We have Acrosonic’s May output expressed as a matrix:

What is the size (order) of the matrix P?

Find a24 (the entry in row 2 and column 4 of the matrix P) and give an interpretation of this number.

Find the sum of the entries that make up row 1 of P and interpret the result.

Find the sum of the entries that make up column 4 of P and interpret the result.

Equality of Matrices

Two matrices are equal if they have the same size and their corresponding entries are equal.

Example - Solve the following matrix equation for x, y, and z

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