Review Intro to Chapter 5: Systems of Linear Equations and Inequalities

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Review Intro to Chapter 5 Systems of Linear
Equations and Inequalities
(p276) Real world problems often involve solving
thousands of equations, sometimes containing a
million variables. Problems ranging from
scheduling airline flights to controlling traffic
flow to routing phone calls over the nations
communication network often require solutions in
a matter of moments. ATTs domestic long
distance network involves 800,000 variables!
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Review Intro to Chapter 5 Systems of Linear
Equations and Inequalities
(p276) Meteorologist describing atmospheric
conditions surrounding a hurricane must solve
problems involving thousands of equations rapidly
and efficiently. The difference between a 2-hour
warning and a 2-day warning is a life and death
issue for thousands of people in the path of one
of natures most destructive forces.
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Review Intro to Chapter 5 Systems of Linear
Equations and Inequalities

• Although we will not be solving 800,000 equations
  with 800,000 variables, we will look at two
  equations with two variables such as
• 2x 3y 6
• 2x y -2

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5.2 Solving Systems of Linear Equations by
Substitution Objectives (p285)

• Solve linear systems by the substitution method.
• Use the substitution method to identify systems
  with no solutions or infinitely many solutions.
• Solve problems using the substitution method.

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5.2 Review Vocabulary (5.1 p276)

• Linear equations All equations that can be
  written in the form
• Ax By C(they are straight lines when
  graphed.)
• A system of linear equations or a linear system
  Two or more linear equations looked at together.

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5.2 Review Vocabulary (5.1 p276) continued

• A solution to a system of linear equations an
  ordered pair that satisfies (makes true) ALL
  equations in the system!Example (-2, 5) is
  solution to the system x y 3 x y
  -7 WHY???
• We can also write the solution is x -2 and y
  5

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5.2 More Reminders from 5.1(p277)

• A system of linear equations can have
• Exactly one solution or
• No solution or
• Infinitely many solutions.
• (just like one equation with one variable!)

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5.2 Last Reminder from 5.1(p277)

• Consider the system
• x 2y 2
• x 2y 6
• The coordinates of the point of intersection give
  the systems solution

All solutions to x 2y 2
All solutions to x 2y 6
Figure 5.1
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5.2 Obj 1 Eliminating a Variable Using the
Substitution Method (p285)

• How can we find the solution to
• x 2y 2
• x 2y 6
• without having to get the graph?
• We use either one of the equations to eliminate
  one of the variables in the OTHER equation.

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5.2 Obj 1 Eliminating a Variable Using the
Substitution Method (p285)

• Solve either of the equations for one variable in
  terms of the other. (If one of the variables is
  already in this form you can skip this step.)
• Substitute the expression found in step 1 into
  the other equation. This will result in an
  equation in one variable.
• Solve the equation obtained in step 2.

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5.2 Obj 1 Eliminating a Variable Using the
Substitution Method (p285)

• Back-substitute the value found in step 3 into
  the equation from step 1. Simplify and find the
  value of the remaining variable.
• Check the proposed solution in BOTH of the
  ORIGINAL given equations.

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5.2 Obj 1 Eliminating a Variable Using the
Substitution Method (p286)

• Solve the system by the substitution method x
  2y 2
• x 2y 6
• (Use the study tip on p. 286 in step 1, if
  possible, solve for a variable whose coefficient
  is 1 or 1 to avoid working with fractions.)
• Return to Process for solving systems by
  substitution

(For more practice try p 285-6 Example 1 check
point 1)
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5.2 Obj 1 Eliminating a Variable Using the
Substitution Method (p286)

• Try p 290 20.
• Directions Solve each system by the
  substitution method. If there is no solution or
  an infinite number of solutions, so state. 2x
  5y 1 -x 6y 8
• Return to procedure slide.
• Remember the study tip p 286
• (For more practice try p 287 Example 3 and check
  point 3)

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5.2 Obj 2 Using the Substitution Method on an
Inconsistent System

• What is an inconsistent system?
• Try p 290 14. (first without a calculator, then
  with!)
• Directions Solve each system by the
  substitution method. If there is no solution or
  an infinite number of solutions, so state. 6x
  2y 7 y 2 3x
• Return to procedure slide.
• What should be true about the graph of this
  system?
• (For more practice try p 288 Example 3 and check
  point 3)

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5.2 Obj 2 Using the Substitution Method on a
System with Infinitely Many Solutions.

• Try p 290 16. (first without a calculator, then
  with!)
• Directions Solve each system by the
  substitution method. If there is no solution or
  an infinite number of solutions, so state. 9x -
  3y 12
• y 3x 4
• Return to procedure slide.
• What should be true about the graph of this
  system?
• (For more practice try p 288 Example 4 and check
  point 4)

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5.2 Obj 3 Solve Systems of Linear Equations
by Substitution Method.

• As the price of a product increases, the demand
  for it decreases, but suppliers are willing to
  produce more!
• Try p 291 34. The weekly demand and supply
  models for a particular brand of scientific
  calculator for a chain of stores are given by the
  Demand model N -53p 1600 and the Supply
  model N 75p 320. In these models, p is the
  price of the calculator and N is the number of
  calculators sold (in the demand model) or
  supplied each week to the stores (in the supply
  model).

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5.2 Obj 3 Solve Systems of Linear Equations
by Substitution Method.

• p 291 34. Continued.
• Demand model N -53p 1600
• Supply model N 75p 320
• Let N represent.
• Let p represent.

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5.2 Obj 3 Solve Systems of Linear Equations
by Substitution Method.

• p 291 34. Continued.
• How many calculators can be sold and supplied at
  12 per calculator?
• Find the price at which supply and demand are
  equal. At this price, how many calculators of
  this type can be supplied and sold each week?
• Return to procedure slide.
• (For more practice try p 281-282 Example 5 and
  check point 5)

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On a sheet of paper for one Quiz point Print your
name and
From TODAYS lesson 1) Describe one main math
idea 2) Identify one math concept or idea that
interest you. 3) Ask at least one question about
the math we covered.