Ch 1 Linear Equations

1
Ch 1Linear Equations

• 1.1 Systems of Linear Equations
• 1.2 Gaussian Elimination
• 1.3 Applications
• 1.4 Chapter Summary

2
1.1 Systems of Linear Equations (Study Book
p 30 Textbook Larson, Ch 1.1)

• Linear equations have form
• a1 x1 a2 x2 a3 x3 an xn b
• with variables xi coefficients ai .
• They arise in many fields of application where
  quantities must satisfy certain constraints or
  demands.

3

• A linear system of m equations in n variables
• has form
• a11 x1 a12 x2 a13 x3 a1n xn
  b1
• a21 x1 a22 x2 a23 x3 a2n xn
  b2
• .
• .
• .
• am1 x1 am2 x2 am3 x3 amn xn
  bm
• A solution to the system is any set of numbers
• s1, s2, , sn,
• that satisfies all the equations,
• when substituted for x1 , x2 , xn ,
  respectively.

4

• Thinking geometrically
• Solutions to a linear equation in 2 variables in
  R2 ax by c
• are the points (x,y) that lie on that straight
  line.
• Solutions to the following equation in R3
• ax by cz d
• are the points (x,y,z) that lie on that plane.

5

• The solutions to a set of such equations,
• (called a linear system)
• are the points that lie on all those lines or
  planes,
• ie their intersection.
• How many solutions (pts of intersection) can we
  get from
• 2 lines? 3 lines?
• 2 planes? 3 planes?
• We can prove generally that linear systems always
  yield
• no solutions at all
• or exactly one solution
• or infinitely many solutions.

6

• Systems with no solutions are called
  Inconsistent.
• Systems that have solutions are called
  Consistent.
• They have exactly one solution or infinitely
  many!
• The types of manipulations we can use to reduce
  eqns without spoiling their solutions, boil down
  to the following so-called
• Elementary Operations
• Interchange two equations.
• Multiply an eqn by a non-zero constant.
• Add a multiple of one eqn to another.

7

• We can use the elementary ops to systematically
• replace the original system with a simpler
• system with the same solutions
• We might aim at triangular form eg
• x - 2y 3z 9 x - 2y 3z
  9
• -x 3y - 4
  y 3z 5
• 2x - 5y 5z 17 z
  2
• so that the solutions pop out by
  back-substitution
• z 2, then y -1 , then x
  1.
• Or we can use the elementary operations to reduce
• further to diagonal form.

8

• Ie, we can systematically reduce the given system
  to
• triangular form, using Gaussian Elimination,
• or diagonal form using Gauss-Jordan Elimination.
  
• The system above yields a single value for each
  variable, ie a unique solution.
• Geometrically, those 3 planes intersect in that
• single point ( 1,-1, 2 ) .
• Some systems have
• infinitely many solutions, or none at all.
• Study Larson Ch 1.1, Ex 8 and 9, carefully,
• to see how this becomes clear.

9

• Note that after row-reduction, the last line (or
  an earlier line) often reveals the nature of the
  solutions.
• eg 0 -2 (or any constant) yields no
  solutions.
• It is a contradiction that no x,y,z can
  satisfy.
• But 0 0 means 0z 0, which is true for any
  z
• In that case, z is a free variable, or parameter.
• Back-subst may yield x y in terms of z (or z
  t ).
• This gives parametric equations for x, y z .
  See Ex 9.
• Each value of t gives a solution (x,y,z)
• ie there are infinitely many solutions.

10
Homework

• Larson Edwards Ch 1.1.
• Note Do these in matrix form, rather,
• as shown in the next section.
• Do odd numbers 1-25, 33-37, 43.
• Do Ed 4 51-59, 73 or Ed 5 51-64, 77.
• Write out careful full solutions to
• Ed 4 Q 3, 7, 14, 27, 37, 51, 55
• or Ed 5 Q 3, 7, 14, 27, 37, 55, 59.

11
Objectives

• Know
• how many solutions a linear system can have
• what consistent inconsistent systems are
• the 3 elementary row operations
• how row operations can be used to reduce a system
  to triangular or diagonal form
• how to back-substitute.