Chapter 2 Simultaneous Linear Equations

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Chapter 2Simultaneous Linear Equations
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2.1 Linear systems

• A system of m linear equations in n variables is
  a set of m equations, each of which is linear in
  the same n variables
• A solution is a set of scalars x1 , x2 , , xn
  that when substituted in the system satisfies the
  given equations.
• A linear system can possess exactly one solution,
  an infinite number of solutions, or no solution.
• A linear system is called consistent if it has at
  least one solution and inconsistent if it has no
  solution.
• A linear system can be written in matrix form Ax
  b (see details on the board)
• A linear system is called homogenous if b0

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2.1 Linear systems (HW example) Modeling a
real-life situation as a linear model

• A manufacturer produces desks and bookcases.
• Desks d require 5 hours of cutting time and 10
  hours of assembling time.
• Bookcases b require 15 minutes of cutting time
  and one hour of assembling time.
• Each day, the manufacturer has available 200
  hours for cutting and 500 hours for assembling.
• The manufacturer wants to know how many desks and
  bookcases should be scheduled for completion each
  day to utilize all available workpower.
• Show that this problem is equivalent to solving
  two equations in the two unknowns d and b .

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2.2 Solutions by Substitution

• Take the first equation and solve for x1 in terms
  of x2 , , xn and then substitute the value of
  x1 into all the other equations, thus eliminating
  it from those equations. This new form is the
  first derived set.
• Working with the first derived set, solve the
  second equation for x2 in terms of x3 , , xn and
  then substitute this value of x2 into the third,
  fourth, etc. equations, thus eliminating it.
• Do this process recursively with other variables.
• The resulting system can be solved by back
  substitution.
• An example on the board.
• We will consider in more details more advanced
  methods which use matrices.

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Augmented matrix of a linear system

• The matrix derived from the coefficients and
  constant terms of a system of linear equations is
  called the augmented matrix of the system.
• The matrix containing only the coefficients of
  the system is called the coefficient matrix of
  the system.
• System Augmented Matrix
  Coefficient Matrix

const.
y
z
x
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Elementary row operations

• (E1) Interchange any two rows.
• (E2) Multiply any row by a nonzero scalar.
• (E3) Add a multiple of a row to another row.
• Two matrices are said to be row-equivalent
• if one can be obtained from the other by a finite
  sequence of elementary row operations.
• Row-equivalent systems have the same set of
  solutions.

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Gaussian Elimination

• Write the augmented matrix of the system.
• Use elementary row operations to transform it to
  an equivalent row-reduced form.
• (this is most often accomplished by using (E3)
  with each diagonal element to create zeros in all
  columns directly below it, beginning with the
  first column)
• The system associated with row-reduced matrix can
  be solved easily by back-substitution.

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Gaussian Elimination Example 1

• Linear System Associated Augmented matrix

R2R1?R2
?(?2)
R3R2?R3
0.5R3?R3
?0.5
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Gaussian Elimination Example 2

• A system with no solution
• Solve the system

0 ?2 ???The original system of linear
equationsis inconsistent.