SECTION 3.1 –SYSTEMS OF LINEAR EQUATIONS, A FIRST LOOK
Definition (Linear Equation, Linear System, Solution, Solution Set, Equivalent Systems)
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Substitution Method
Example 1
Find the solution to the following systems.
x − 2y = 4
1.)
3x + y = 5
2.)
6c − 2d = 4
7c + d = 13
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3.)
2a − 5b + c = 0
6a + 4b − 3c = −12
2a + 4c = 6
Elementary Equation Operations
1.)
Replacement: Replace one equation by the sum of itself and a multiple of another equation.
2.)
Interchange: Interchange two equations.
3.)
Scaling: Multiply both sides of an equation by the same nonzero constant.
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Elimination Method
Example 2
Find the solution to the following systems.
9 x − 2y = 5
1.)
3 x − 3 y = 11
2.)
− 3m + n = 2
7m − 8n = 1
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3.)
2r + 4s − t = −7
− 5r + 2s + t = −5
− r + 4 s = −6
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Example 3
Find the solution to the following systems.
2s + 4t = 8
1.)
3s + 6 t = 3
2.)
− 3 x + 8 y = 29
− 6 x + 16 y = 58
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Example 4
A collection of 42 coins consists of dimes and nickels. The total value is $3. How many dimes
and how many nickels are there?
Existence & Uniqueness
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SECTION 3.2 – LARGER SYSTEMS: MATRIX REPRESENTATION & GAUSS-JORDAN
ELIMINATION
Definition (Matrix, Element, Matrix Size)
Definition (Coefficient Matrix, Augmented Matrix)
Example 1
Find the coefficient and augmented matrices of each of the following systems. Also, identify the
size of each matrix.
System
Coefficient Matrix
Augmented Matrix
x − 2y = 4
3x + y = 5
x − 2y + 3z = 0
2x − y − z = −1
x+ y−z =1
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Definition (Leading Entry)
1.)
Echelon Form
Row Reduced Echelon Form
All nonzero rows are above any rows of
1.) All nonzero rows are above any rows of all
all zeros.
2.)
3.)
zeros.
Each leading entry of a row is in a column
2.) Each leading entry of a row is in a column
to the right of the leading entry of the row
to the right of the leading entry of the row
above it.
above it.
All entries in a column below a leading
3.) All entries in a column below a leading
entry are zeros.
entry are zeros.
4.) The leading entry in each nonzero row is 1.
5.) Each leading 1 is the only nonzero entry in
its column.
Example 2
Determine if each of these is an echelon matrix, a row reduced echelon matrix, or neither.
 1 4 7


2 5 8
3 6 9


 1 2 2 0


 0 5 4 1
0 0 0 0


 1 0 0 0


0 0 0 0
0 0 0 0


 92 65 


 0 12 
0 0


 1 0 0


0 1 0
 0 0 1


6

0
0

0

5 2

8 1
0 5

0 2 
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Elementary Row Operations
1.)
Replacement: Replace one row by the sum of itself and a multiple of another row.
2.)
Interchange: Interchange two rows.
3.)
Scaling: Multiply all entries of a row by a nonzero constant.
Note
Replacement and scaling operations may be combined into more efficient row operations of the
form aR n + bR m → R n where m ≠ n and a ≠ 0 .
Definition (Row Reduction)
The Row Reduction Algorithm
1.) Begin with the leftmost nonzero column. Interchange rows, if necessary, so that the
leading entry for the top row is in this column.
2.) Use row replacement operations to create zeros in all positions below this leading entry.
3.) Cover the row and column containing this leading entry (and any rows above it or columns
to the left of it) and apply steps 1-3 to the submatrix that remains. Repeat this process until
there are no more nonzero rows to modify.
*Your matrix should now be in echelon form.*
4.) Beginning with the rightmost leading entry and working upward and to the left, create zeros
above each leading entry. If the leading entry is not 1, use a scaling operation to make it a
1.
*Your matrix should now be in row reduced echelon form.*
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Example 3
Row reduce each of these matrices into row reduced echelon form.
1 4 0 7 


 2 7 0 10 
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 3 −4 −5 8 


0 
 1 −2 1
− 4 5
9 − 9 

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Definition (Equivalent Matrices)
Using Row Reduction To Solve A Linear System
1.) Write the augmented matrix for the system.
2.) Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form.
Decide whether the system is consistent. If there is no solution, stop; otherwise continue.
3.) Option 1: Rewrite the new augmented matrix as a system and use substitution to find the
solution.
Option 2: Continue row reduction to obtain the reduced echelon form. Write the system of
equations corresponding to the row reduced echelon matrix.
Note
If a leading entry of any row is in the right most column of any matrix that is equivalent to the
augmented matrix for a system, then the system is inconsistent. It is most convenient to look
only at the leading entries of the echelon form of the augmented matrix to determine if a system
is consistent.
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Example 4
Find solutions to the following systems using matrices.
3 x + 4 y = 12
3x − 8y = 0
6a + 3b = 9
8a + 4b = −2
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3r + 2s + 2t = 29
9r + 8s + 9t = 116
r + 2s + 9t = 86
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SECTION 3.3 – MORE MODELING AND APPLICATIONS
Example 1
Grace sells two kinds of granola. One is worth $4.05 per pound and the other is worth $2.70 per
pound. She wants to blend the two granolas to get a 15 lb mixture worth $3.15 per pound. How
much of each kind of granola should be used?
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Example 2
Animals in an experiment are to be kept under a strict diet. Each animal is to receive, among
other things, 20 grams of protein and 6 grams of fat. The laboratory technician is able to
purchase two food mixes. Mix A contains 10% protein and 6% fat and Mix B contains 20%
protein and 2% fat. How many grams of each mix should be used to obtain the right diet for a
single animal?
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Example 3
A 480 m wire is cut into three pieces. The second piece is three times as long as the first. The
third is four times as long as the second. How long is each piece?
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Example 4
A furniture manufacturer makes chairs, coffee tables, and dining room tables. Each chair
requires 10 minutes of sanding, 6 minutes of staining, and 12 minutes of varnishing. Each
coffee table requires 12 minutes of sanding, 8 minutes of staining, and 12 minutes of varnishing.
Each dining room table requires 15 minutes of sanding, 12 minutes of staining, and 18 minutes
of varnishing. The sanding bench is available for 16 hours per week, the staining bench is open
for 11 hours per week, and the varnishing bench is available for 18 hours per week. How many
(per week) of each type of furniture should be made so that the benches are fully utilized?
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Example 5
The sum of the digits in a four-digit number is 10. Twice the sum of the thousands digit and the
tens digit is 1 less than the sum of the other two digits. The tens digit is twice the thousands
digit. The ones digit equals the sum of the thousands digit and the hundreds digit. Find this fourdigit number.
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Example 6
A chemist has two saline solutions: one has 10% concentration of saline and the other has 35%
concentration of saline. How many cubic centimeters of each solution should be mixed together
in order to obtain 60 cubic centimeters of solution with a 25% concentration of saline?
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Example 7
A horticulturist wishes to mix three types of fertilizer. Type A contains 25% nitrogen, Type B
contains 35% nitrogen, and Type C contains 40% nitrogen. She wants a mixture of 4000
pounds with a final concentration of 35 58 % nitrogen. The final mixture should also contain three
times as much of Type C than of Type A. How much of each type is in the final mixture?
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SECTION 3.4 – OTHER APPLICATIONS INVOLVING MATRICES
Scalar Multiplication
Matrix Addition & Subtraction
Additional Properties
Let A, B, and C be matrices of the same size, and let r & s be scalars.
1.) A + B = B + A
2.)
(A + B ) + C = A + (B + C)
Example 1
 6

Let A =  − 2
 1

3.) A + 0 = A
5.)
4.) r (A + B ) = rA + rB
6.)
(r + s)A = rA + sA
r (sA ) = (rs)A
4
 8 − 3



3  and B =  5
1 . Compute each of the following:
− 2
0 
4 

- 2A =
B - 2A =
A + 2B =
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Example 2
Ms. Smith and Mr. Jones are salespeople in a new-car agency that sells only two models.
August was the last month for this year’s models and next year’s models were introduced in
September. Gross dollar sales for each month are given in the following matrices:
August Sales
Compact
Ms. Smith
Mr. Jones
September Sales
Luxury
 $54,000 $88,000 
=A

$0 
 $126,000
Compact
Ms. Smith
Mr. Jones
Luxury
 $228,000 $368,000 
 = B

 $304,000 $322,000 
a.) What were the combined dollar sales in August and September for each salesperson and
each model?
b.) What was the increase in dollar sales from August to September?
c.) If both salespeople receive 5% commissions on gross dollar sales, compute the commission
for each person for each model sold in September.
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Inner Product (aka Dot Product)
Example 3
Compute each of the following.
 − 5
 
(2 − 3 0) •  2  =
 − 2
 
 2
 
3
(− 1 0 3 2) •   =
4
 
 − 1
 
Matrix Multiplication
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Example 4
0 − 1
1
 2
 7 −5
 1
 , B = 
 , C = 
Let A = 
2
 4 −5
 1 − 4 − 3
− 2
Compute each of the following:
2
 3
 , and D = 
1
−1
5
.
4 
AC =
CA =
CD =
CB =
WARNINGS!
1.) In general, AB ≠ BA .
2.) The cancellation laws do NOT hold for matrix multiplication. That is, if AB = AC then it is
NOT necessarily true that B = C .
3.) If a product AB is the zero matrix, you CANNOT conclude in general that either A = 0 or
B = 0.
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Example 5
A nutritionist for a cereal company blends two cereals in three different mixes. The amounts of
protein, carbohydrate, and fat (in grams per ounce) in each cereal are given by matrix M. The
amounts of each cereal used in three mixes are given by matrix N.
Cereal A
Protein
Carbohydrate
Fat
Cereal B
 4 g / oz 2 g / oz 


 20 g / oz 16 g / oz  = M


1 g / oz 
 3 g / oz
Mix X
Cereal A
Cereal B
Mix Y
Mix Z
15 oz 10 oz 5 oz 


 5 oz 10 oz 15 oz  = N


a.) Find the amount of protein in mix X.
b.) Find the amount of fat in mix Z.
c.) Discuss possible interpretations of the elements in the matrix products MN and NM.
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