1 | Page Hannah Province – Mathematics Department – Southwest Tennessee Community College 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.
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1 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College
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.
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Example - Find the solution to the following system of equations graphically.
1.
189
96
yx
yx
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2.
6
3055
xy
yx
3.
1236
12
yx
yx
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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.
1.
1223
12
yx
yx
2.
336
12
yx
yx
3.
102
1224
xy
yx
5 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College
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.
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2.
1224
135
yx
yx
3.
20105
2144
xy
yx
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Section 7.3 – Matrices
A matrix is a rectangular array of elements.
o 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:
2 4 6 22
3 8 5 27
2 2
x y z
x y z
x y z
2 4 6 22
3 8 5 27
2 2
x y z
x y z
x y z
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Example -
Example -
Write the augmented matrix of the given system of equations.
3x+4y=7
4x-2y=5
2 1 2
Write the system of equations corresponding to this
augmented matrix. Then perform the row operation
on the given augmented matrix. R 4
1 3 3 5
4 5 3 5
3 2 4 6
r r
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10 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College
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 –
Find , given
3 5 2 6 4 and
4 1 0 3 7
A B
A B
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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.
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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.
o 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 to the 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.
1 1 1
2 2 2
a b c
a b c
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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.
o 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 the following equations.
x + 2y = 16
y = 7
Substituting 7 for y in the equation, then solving for x, x = 2.
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A matrix with 1s down the main diagonal and 0s below the
1s is said to be in row-echelon form. We use row operations
on the augmented matrix. These row operations are just like
what you did when solving a linear system by the addition method.
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Example -
2 1 2
Write the system of equations corresponding to this
augmented matrix. Then perform the row operation
on the given augmented matrix. R 4
1 3 3 5
4 5 3 5
3 2 4 6
r r
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Gauss-Jordan Elimination
Using Gaussian elimination we obtain a matrix in row-echelon form,
with 1s down the main diagonal and 0s below the 1s. When you have
accomplished this you must back-substitute to get your answers.A second
method called Gauss-Jordan elimination continues the process until a matrix
with 1s down the main diagonal and 0s in every position above and below each
1 is found. Such a matrix is said to be in reduced row-echelon form. For a
system in three variables, x,y, and z, we must get the augmented matrix into
the form seen below.
Sometimes it is advantageous to write a matrix in
reduced row echelon form. In this form, row operations
are used to obtain entries that are 0 above as well as
below the leading 1 in a row. The advantage is that the
solution is readily found without needing to back-substitute.
There will be a second advantage in the future when we
discuss the inverse of a matrix.
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Example - Solve the following system of equations by using matrices.
x + 2y = 16
2x + y = 11
18 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College
Graphing Calculator-Matrices, Reduced Row Echelon Form
1To work with matrices we need to press 2nd x keys
to get the Matrix Menu.
To type in a matrix, cursor to the right twice to getto EDIT,
press ENTER and type in the dimensions of matrix A. Press
ENTER after each number, then type in the numbers that
comprise the matrix, again pressing ENTER after each number.
To get out of the matrix menu press QUIT (2nd MODE).
1
To get the reduced row-echelon form bring up
the Matrix Menu (2nd x ) and cursor to MATH.
Cursor down to B, rref (reduced row echelon form).
Press ENTER. (Just row echelon form is "A:ref.")
To type the name of the matrix again bring up
the Matrix Menu and under NAMES, choose
#1, press ENTER, press ENTER again any
you will have your new matrix.
A
19 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College
Example -
A System of Equations with an Infinite Number of Solutions
Example - Solve the system of equations given by
Solve this system of equations using matrices (row operations).
3 5 3
15 5 21
x y
x y
2 3 2
3 2 1
2 3 5 3
x y z
x y z
x y z
2 3 2
3 2 1
2 3 5 3
x y z
x y z
x y z
20 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College
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:
MMooddeell AA MMooddeell BB MMooddeell CC MMooddeell DD
LLooccaattiioonn II 332200 228800 446600 228800
LLooccaattiioonn IIII 448800 336600 558800 00
LLooccaattiioonn IIIIII 554400 442200 220000 888800
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:
1
3 4
5 5 1
x y z
x y z
x y z
1
3 4
5 5 1
x y z
x y z
x y z
320 280 460 280
480 360 580 0
540 420 200 880
320 280 460 280
480 360 580 0
540 420 200 880
320 280 460 280
480 360 580 0
540 420 200 880
P
320 280 460 280
480 360 580 0
540 420 200 880
P
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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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