Intermediate Algebra Chapter 5

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					 Intermediate Algebra
       Chapter 4
     • Systems
        • of
• Linear Equations
          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.
      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
    Independent Equations

• Equations with
  different graphs.
          Algebraic Check
• Same Line

        a1 x  b1 y  c1
        a2 x  b2 y  c2

     a1   b1   c1
            
     a2   b2   c2
           Algebraic Check
• Parallel Lines

 a1 x  b1 y  c1
 a2 x  b2 y  c2
                             
 a1   b1   c1
        
 a2   b2   c2
           Algebraic Check
• Meet in a point {(x,y)}
      a1 x  b1 y  c1
      a2 x  b2 y  c2
   a1   b1   c1
          
   a2   b2   c2
 Calculator Method for Systems

• Solve each equation for y
• Input each equation into Y=
• Graph
• Set Window
• Use CalIntersect
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
        Objective

• Solve a System of
  Equations using the
  Substitution Method.
        Substitution Method
• 1. Solve one equation for one variable
• 2. In other equation, substitute the
  expression found in step 1 for that
  variable.
• 3. Solve this new equation (1 variable)
• 4. Substitute solution in either original
  equation
• 5. Check solution in original equation.
 Althea Gibson – tennis player

• “No matter what
  accomplishments you
  make, someone helped
  you.”
Intermediate Algebra

     •The
 •Elimination
   •Method
  Notes on elimination method
• Sometimes called addition
  method
• Goal is to eliminate on of the
  variables in a system of equations
  by adding the two equations, with
  the result being a linear equation in
  one variable.
• 1.Write both equations in ax +
  by = c form
• 2. If necessary, multiply one or
  both of the equations by
  appropriate numbers so that the
  coefficients of one of the
  variables are opposites.
    Procedure for addition method cont.
• 3. Add the equations to eliminate a
  variable.
• 4. Solve the resulting equation
• 5. Substitute that value in either of
  the original equations and solve for
  the other variable.
• 6. Check the solution.
    Procedure for addition method cont.
• Solution could be ordered pair.
• If a false statement results i.e. 1 =
  0, then lines are parallel and
  solution set is empty set.
  (inconsistent)
• If a true statement results i.e. 0 = 0,
  then lines are same and solution set
  is the line itself. (dependent)
          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)}
  Practice Problem


2x  y  1
2 x  y  3
   Practice problem


3x  4 y  5
9 x  12 y  15
Special Note on Addition Method

• Having solved for one variable,
  one can eliminate the other
  variable rather than substitute.
• Useful with fractions as
  answers.
Practice Problem – eliminate one
   variable and than the other
    3    3    15
      x   y
    4    4     4
    4    5
      x   y3
    3    3
• Answer: {(8/3,1/3)}
         Confucius

• “It is better to light one
  small candle than to
  curse the darkness.”
Intermediate Algebra 4.2

      • Systems
         • Of
     • Equations
         • In
  • Three Variables
         Objective

• To use algebraic
  methods to solve linear
  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) 3x  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.”
         Cramer’s Rule

• Objective: Evaluate
  determinants of 2 x 2 matrices
• Objective: Solve systems of
  equations using Cramer’s Rule
         Determinant
       a b 
If A       then det[ A] 
       c d 
   a b
        ad  bc
   c d
       Cramer’s rule intuitive
• Each denominator, D is the
  determinant of a matrix containing
  only the coefficients in the system. To
  find D with respect to x, we replace the
  column of s-coefficients in the
  coefficient matrix with the constants
  form the system. To find D with
  respect to y, replace the column of y-
  coefficients in the coefficient matrix sit
  the constant terms.
   Sample Problem: Evaluate:


           3 2
           2    4
• Answer = 16
 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.”
    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.
Distance problem equations

    0.25( x  y )  1
    0.1( x  y )  1
     x y 4
     x  y  10
          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.”

				
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posted:9/13/2012
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