Mathematics 116 Chapter 5
Bittinger
•Linear Systems
•and
•Matrices
Mark Twain - American author (1835-1910)
• “What is the most rigorous law
of our being? Growth. No
smallest atom of our moral,
mental, or physical structure
can stand still a year. It grows,
it must grow…nothing can
prevent it.”
Objective
• Determine if an ordered
pair is a solution for a
system of equations.
System of Equations
• Two or more equations
considered simultaneously form
a system of equations.
a1 x  b1 y  c1
a2 x  b2 y  c2
Checking a solution to a system
of equations
• 1. Replace each variable
in each equation with its
corresponding value.
• 2. Verify that each
equation is true.
Solving systems of equations
• Solve Graphically
• Graph both equations with appropriate
window.
• Determine point of intersection using
intersect feature of calculator
• Graph by hand – more points, more care,
the more accurate
Graphical Analysis
3 Possibilities
• Two lines meet in a point
• Two lines are parallel
• Two lines are the same
Meet in a Point
• Consistent – has a single solution at the
point of intersection
• Independent – graphs are different and
intersect at one point.
• They have different slopes.
• Solution is an ordered pair
a2

b2

c2
Parallel Lines
• Inconsistent system – the system has no
solution.
  or 
• Solution set
• Independent – the graphs are different
• Lines are parallel
• Slopes are the same
• y intercept is different
Same Line
• Consistent System – the system has an
infinite number of solutions
• Dependent – the graphs are identical
• Have the same slope
• Have the same y intercept
Graphing Procedure
• 1. Graph both equations in the
same coordinate system.
• 2. Determine the point of
intersection of the two graphs.
• 3. This point represents the
estimated solution of the system of
equations.
Graphing observations
• Solution is an estimate
• Lines appearing parallel have to
be checked algebraically.
• Lines appearing to be the same
have to be checked
algebraically.
Classifying Systems
• Meet in Point – Consistent –
independent
• Parallel – Inconsistent –
Independent
• Same – Consistent - Dependent
Def: Dependent Equations
• Equations with
identical graphs
Def: Dependent Equations
• Equations with
identical graphs
Calculator Problem
y  2 x  3
x  2y  4
 2, 1
Calculator Problem 2
3x  4 y  8
3
y   x3
4

Calculator Problem 3
6x  2 y  4
y  3x  2
 x, y  | y  3x  2
Algebraic Analysis
a1 x  b1 y  c1
a2 x  b2 y  c2
Same line
a1 x  b1 y  c1
a2 x  b2 y  c2
a1 b1 c1
 
a2 b2 c2
Same line example
• 2x + 3y = 4
• 4x + 6y = 8
• Solution set
 x, y  | 2x 3 y  4
Parallel Lines
a1 b1 c1
 
a2 b2 c2
Parallel lines example
• 2x + 3y = 4
• 4x + 6y = 5
Solve using substitution
• Isolate one of the variables
• In the other equation, substitute the
expression
• Solve the new equation for one
unknown
• Substitute the value obtained and
solve for the other variable
• Check the result
Substitution: Special Notes For
Lines
• solution is ordered pair - Two lines
• Obtain false statement such as 0 = 1
• ----parallel lines – solution set is empty
set
• Obtain true statement – such as 0 = 0
• This is the same line and the solution
set is the line itself.
Objective
• Use systems of equations to
model and solve real-life
problems.
Break Even
• C = Total Cost = cost per unit *number of
units +initial cost
• R = Total Revenue = Price per unit * # of
units
• Break Even is R = C
Theodore Roosevelt
• “I think we consider too
much the good luck of the
early bird and not enough the
bad luck of the early worm.”
Mathematics 116
• Systems of Linear Equations
• In
• Two Variables
Solve by Elimination
• Write in standard form
• Clear equations of fractions or decimals
• Multiply one or both equation by number(s)
so that a pair of terms are additive inverses
• Add the equations
• Solve for one unknown
• Substitute to find other unknown
• Check
Elimination: Special Notes
• solution is ordered pair - Two lines
• Obtain false statement such as 0 = 1
• ----parallel lines – solution set is empty
set
• Obtain true statement – such as 0 = 0
• This is the same line and the solution
set is the line itself
Practice Problem
x y 6
2 x  5 y  16
• Answer {(2,4)}
Practice Problem Hint: eliminate
x first
4 x  3 y  2
6 x  7 y  7
• Answer {(-7/2,-4)}
Dale Earnhardt
• “You win some, you lose
some, you wreck some.”
Confucius
• “It is better to light one
small candle than to
curse the darkness.”
College Algebra
• Systems
• Of
• Equations
• In
• Three Variables
Def: linear equation in 3
variables
• is any equation that can be
written in the standard form
ax + by +cz =d where a,b,c,d
are real numbers and a,b,c
are not all zero.
Def: Solution of linear equation
in three variables
• is an ordered triple
(x,y,z) of numbers that
satisfies the equation.
Procedure for 3 equations, 3
unknowns
• 1. Write each equation in the form
ax +by +cz=d
• Check each equation is written
correctly.
• Write so each term is in line with a
corresponding term
• Number each equation
Procedure continued:
• 2. Eliminate one variable from
one pair of equations using the
elimination method.
• 3. Eliminate the same variable
from another pair of
equations.
• Number these equations
Procedure continued
• 4. Use the two new equations
to eliminate a variable and solve
the system.
• 5. Obtain third variable by back
substitution in one of original
equations
Procedure continued
• Check the ordered
triple in all three of the
original equations.
Sample problem 3 equations
(1)
x y z 2
(2) 2 x  y  2 z  1
(3) 3 x  2 y  z  1
Answer to 3 eqs-3unknowns
•{(-2,3,1)}
Bertrand Russell – mathematician
(1872-1970)
• “Mathematics takes us still
further from what is human,
into the region of absolute
necessity, to which not only the
actual world, but every possible
world, must conform.”
Intermediate Algebra 5.5
• Applications
• Objective: Solve application
problems using 2 x 2 and 3 x 3
systems.
Mixture Problems
• ****Use table or chart
• Include all units
• Look back to test
reasonableness of answer.
Sample Problem
• How many milliliters of a
10% HCl solution and
30% HCl solution must
be mixed together to
make 200 milliliters of
15% HCl solution?
Mixture problem equations
x  y  200
0.10 x  0.30 y  30
Mixture problem answers
• 150 mill of 10% sol
• 50 mill of 30% sol
• Gives 200 mill of 15% sol
Distance Problems
• Include Chart and/or picture
• Note distance, rate, and time in
chart
• D = RT and T = D/R and R=D/T
• Include units
• Check reasonableness of answer.
Sample Problem
• To gain strength, a rowing crew
practices in a stream with a fairly quick
current. When rowing against the
stream, the team takes 15 minutes to
row 1 mile, whereas with the stream,
they row the same mile in 6 minutes.
Find the team’s speed in miles per hour
in still water and how much the current
changes its speed.
• Mathematics 116
• 3 equations and 3 unknowns
Sample Problem
• To gain strength, a rowing crew
practices in a stream with a fairly quick
current. When rowing against the
stream, the team takes 15 minutes to
row 1 mile, whereas with the stream,
they row the same mile in 6 minutes.
Find the team’s speed in miles per hour
in still water and how much the current
changes its speed.
Answer
• Team row 7 miles per hour
in still water
• Current changes speed by
3 miles per hour
Joe Paterno – college football
coach
• “The will to win is
important but the will
to prepare is vital.”
Gaussian Elimination
Row Operations
• Interchange two equations
• Multiply one of the equations
by a nonzero constant
• Add a multiple of one equation
to another equation.
Anton Pavlovich Chekhov
• “The problem is that we
attempt to solve the simplest
questions cleverly, thereby
rendering them unusually
complex. One should seek
the simple solution.”
Mathematics 116
• Matrices
•and
•Systems of Equations
Matrix: order m x n
 a11
a
 21


 am1
a12
a22
am 2
a1n 

a2 n



amn 
Matrix Terms
• Double subscript
• Order
• Square matrix
• Main diagonal
• Row matrix
• Column matrix
Augmented Matrix
• A matrix derived from a system of linear
equations each written in standard form
with the constant term on the right.
• The matrix derived from the coefficients of
the system but not including the constant
terms is the coefficient matrix of the system.
Elementary Row Operations
• 1. Interchange two rows.
• 2. Multiply a row by a nonzero constant
• 3. Add a multiple of a row to
another row
Gaussian Elimination with
Augmented Matrix
• Write the augmented matrix of the system
of linear equations
• Use elementary row operations to rewrite
the augmented matrix in row-echelon form.
• Write the system of linear equations
corresponding to the matrix in row echelon
form and use back substitution to find the
solution.
Gauss Jordan
• Do Gaussian Elimination
• Continue until principal diagonal is all 1’s
• Read solution directly
David Cronenberg
• “Everybody’s a mad
scientist, and life is their lab.
We’re all trying to
experiment to find a way to
live, to solve problems, to
fend off madness and chaos.”
Mathematics 116
• Operations
•With
•Matrices
Equality of Matrices
• Two matrices are equal if they have
• (1) the same order
• (2) the corresponding entries are equal.
A   aij   bij   B
Matrix Addition
• Add two matrices of the same
order by adding their
corresponding entries.
A  B   aij  bij 
Matrix addition
• The sum of two matrices of
different order is undefined.
Scalar Multiplication
• If A is an m x n matrix and c is a scalar then
• (c is a real number)
cA  c  aij   caij 
Properties of Matrices
•
•
•
•
•
•
A + B = B + Acommutative for +
A+(B+C)=(A+B)+C
associative for +
(cd)A = c(dA) associative for scalar *
1A = A
scalar identity
c(A + B) = cA + cB distributive
(c + d)A = cA + dA
distributive
Matrix Multiplication
• Multiply row by column
• The entry in the ith row and jth column of
product AB obtained by multiplying the
entries in the ith row of A by the
corresponding entries in the jth column of B
and then adding the results.
Matrix multiplication
• ***The number of columns of the first
matrix must equal the number of rows of
the second matrix.
• The outside two indices give the order of
the product.
Amxn x Bnx p  ABmx p
Matrix multiplication properties
•
•
•
•
A(BC)=(AB)C
A(B+C)=AB+AC
(A+B)C=AC+BC
c(AB)=(cA)B
Associative * matrices
left distributive matrices
right distributive matrices
associative of scalar *
Identity Matrix
• Square matrix
• 1’s on main diagonal
• 0’s elsewhere
• Denoted by I
Objectives
• Decide whether two matrices
are equal
• Add and subtract matrices
• Multiply matrices by a scalar
• Multiply two matrices
• Use matrices to model and
solve real-life problems.
Matrix-Calculators
•
•
•
•
•
2nd MATRIX
Names Math
EDIT
Names are in [ ]
Math – Transform & row operations
EDIT – input matrix – give order and
entities – can be edited.
College Algebra
• The Inverse
• Of
•A
• Square Matrix
Inverse of a Square Matrix
• Let A be an n x n matrix and let I be the n x
n identity matrix. If there exists a matrix A
inverse such that
1
1
AA  I n  A A
1
A is called the inverse of A
Finding Inverse matrices using
matrices
• Put equations in standard form
• Adjoin the identity matrix to the coefficient
matrix of the system
• Apply Gauss Jordan elimination to this
matrix.
 A I    I
A 
1
Quick Method 2 x 2 Inverse
a b 
A

c
d


d

b


1
1
A 


ad  bc  c a 
Inverse Matrix with Calculators
• If A is an invertible matrix, the system of
linear equations represented by AX = B has
a unique solution given by
1
X A B
Objectives
• Verify that two matrices are inverses of each
other
• Use Gauss-Jordan elimination to find
inverses of matrices
• Use the “quick method” to find the inverse
of matrices
• Solve systems of equations using inverse of
matrix method
Lee Iacocca
• “Boys, there ain’t no free lunches in
this country. And don’t go spending
you whole life commiserating that you
got the raw deals. You’ve got to say, I
think that if I keep working at this and
want it bad enough I can have it. It’s
called perseverance.”
College Algebra
•The Determinant
•of
•a
•Square Matrix
Def of 2 x 2 Determinant of 2 x 2
matrix
 a11 a12 
det( A)  det 

 a21 a22 
a11 a12
 A
 a11a22  a21a12
a21 a22
Minors of square matrix
• If A is a square matrix, the minor of an
entity is the determinant of the matrix
obtained by deleting the row and column of
the entity.
Cofactor
i j
Cij  (1) M ij
where M ij is the MINOR
Determinant of Square Matrix
• Expand using coFactors
• Expanding along the first row
A  a11C11  a12C12 
 a1nC1n
Determinant of Matrix
by Expansion by Minors
• Expand by Minors
• Pick a row or Column
• Multiply each entity of selected
row or column by its minor.
• Connect with appropriate sign
• Combine
Dwight Eisenhower
• “No one can defeat us
unless we first defeat
ourselves.”
College Algebra 5.8
• Applications
• of
• Matrices
• and
• Determinants
Objective
• Use Cramer’s Rule to solve
systems of linear equations.
Cramer’s Rule
• Used to solve systems of equations
• Find the determinant of the coefficient
matrix
• Designated by D
• Use the column of constants as
replacements for the coefficients of other
variables to make other Determinants
Cramer’s Rule
a1 x  b1 y  c1
ax x  b2 y  c2
D
a1
b1
a2
b2
Cramer’s Rule Continued
Dx 
c1
b1
c2
b2
Dx
x
D
y
Dy 
Dy
D
a1
c1
a2
c2
Cramer’s Rule Continued
• Cramer’s Rule does not apply
when the determinant of the
coefficient matrix is zero.
• If determinants with respect to x
and y are also zero, then the system
of 2 equations represents the same
line.
Sample Problem: Evaluate:
3 2
2
• Answer = 16
4
Sample Cramer’s Rule problem
• Solve by Cramer’s Rule
2 x  3 y  5
3x  y  9
Cramer’s Rule Answer
D  11
Dx  22
Dy  33
Dx
22
x

2
D
11
Dy
33
y

 3
D
11
Senecca
• “It is not because things
are difficult that we do
not dare, it is because we
do not dare that they are
difficult.”
Objective
• Use determinants to find the
area of triangles.
Area of Triangle
vertices ( x1 , y1 ),( x2 , y2 ),( x3 , y3 )
x1
1
Area   x2
2
x3
y1 1
y2 1
y3 1
Objective
• Use determinants to
decide whether points
are collinear.
Test for Collinear Points
( x1 , y1 ),( x2 , y2 ),( x3 , y3 )
are collinear iff
x1
y1 1
x2
y2 1  0
x3
y3 1
Objective
• Use determinants to determine
the equation of a line given two
points.
Equation of Line
• Given two points, the equation of a line is
(expand using first row)
Given two po int s : ( x1 , y1 ),( x2 , y2 )
x
y
1
x1
y1 1  0
x2
y2 1
Determinant of Square Matrix
• Graphing Calculator
• Use Matrix – Math – det
• Input Matrix A
• det([A])
Cramer’s Rule & Calculator
• Can input and determine
determinants from calculator
matrix menu.
• Can use program Algebra A
• [Cramer’s Rule]
College Algebra
Systems of Equations
• Solve by the following:
• (1) graphing
• (2) substitution
• (3) elimination
Systems continued
Gaussian elimination with
augumented matrix
(5) GaussJordan
(6) Inverse of Matrix
(7) Cramer’s Rule
(8) Pivoting
• (4)
•
•
•
•
Systems solutions & Calculator
•
•
•
•
•
•
•
•
•
(1) Numerical – use table
(2) Graphing – use intersect
(3) Matrix-Math-rref
(4) Use inverse of matrix
(5) Use determinants
(6) Program - ALGEBRA Cramer’s Rule
(7) Program – SIMULT
(8) Program ALG2 – 3:SYSTEM
(9) Program MATH99-ALGEBRA-6SYSTEMS (uses lots of memory)
Systems of Non-linear Equations
•
•
•
•
Graph First
Use Substitution
Use Elimination
Examine graph for reasonableness of
results.
Example: Nonlinear systems
9 x  4 y  20
2
2
x  y 8
2
2
 2,2 ,  2, 2 ,  2,2 ,  2, 2
Example: nonlinear systems
y  x2
x  y 8
2
2
solution
2,2
 
Charles Haddon Spurgeon
• “By perseverance the
snail reached the ark.”