Intro to Chapter 5: Systems of Linear Equations and Inequalities

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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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Intro to Chapter 5 Systems of Linear Equations
and Inequalities S
(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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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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5.1 Solving Systems of Linear Equations by
Graphing Objectives (p276)

• Decide whether an ordered pair is a solution of a
  linear system.
• Solve systems of linear equations by graphing.
• Use graphing to identify systems with no
  solutions or infinitely many solutions.

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5.1 Obj 1 Systems of Linear Equations and
Their Solutions -Vocabulary (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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5.1 Obj 1 Systems of Linear Equations and
Their Solutions-Vocabulary (p276)

• 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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5.1 Obj 1 Systems of Linear Equations and
Their Solutions- (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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5.1 Obj 1 Systems of Linear Equations and
Their Solutions-

• Try p 283 8. (first without a calculator, then
  with!) (work with fractions-reduce-dont change
  to decimals!)
• Directions Determine whether the given ordered
  pair is a solution of the system.

(For more practice try p 277 Example 1 and check
point 1)
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5.1 Obj 2 Solve Systems of Linear Equations by
Graphing. (p278)

• 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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5.1 Obj 2 Solve Systems of Linear Equations by
Graphing. (p278)

• Every point on the blue line makes x 2y 2
  true. (That really means when the coordinates
  of each point on the blue line, replace the
  variables in x 2y 2, we get a true
  statement.)
• Every point on the red line makes x - 2y 6
  true.
• So where they cross is a common solution to both
  equations, the solution to the system. (Return
  to diagram)

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5.1 Obj 2 Solve Systems of Linear Equations by
Graphing.

• Try p 283 18. (first without a calculator, then
  with!)
• Directions Solve each system by graphing. If
  there is no solution or an infinite number of
  solutions, so state. 5x y 10 2x y
  4
• Return to procedure slide.
• (For more practice try p 278 Example 2 and check
  point 2)

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5.1 Obj 2 Solve Systems of Linear Equations by
Graphing.
P279 Discover for Yourself

• Must 2 lines intersect at exactly one point?
  Sketch 2 lines that have less than one
  intersection point. Now sketch 2 lines that have
  more than one intersection point. What does this
  say about each of these systems?

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5.1 Obj 2 Solve Systems of Linear Equations by
Graphing. (p278)

• Graph the first equation.
• Graph the 2nd equation on the same axes.
• If the lines intersect at a point, the
  coordinates of the point is the solution.
• Check the solution in BOTH equations.

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5.1 Obj 2 Solve Systems of Linear Equations
by Graphing.

• Try p 283 20. (first without a calculator, then
  with!) (work with fractions-reduce-dont change
  to decimals!)
• Directions Solve each system by graphing. If
  there is no solution or an infinite number of
  solutions, so state. y x 1 y 3x -
  1Return to procedure slide.
• (For more practice try p 279-80 Example 3 and
  check point 3)

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5.1 Obj 3 Linear Systems Having No solution
or Infiniteely Many Solutions P 280

• The Number of Solutions to a System of Two Linear
  Equations is given by one of the following

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5.1 Obj 3 Solve Systems of Linear Equations
by Graphing.

• Try p 283 26. (first without a calculator, then
  with!)
• Directions Solve each system by graphing. If
  there is no solution or an infinite number of
  solutions, so state. y 3x - 1 y 3x
  2 Return to procedure slide.
• (For more practice try p 281 Example 4 and check
  point 4)

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5.1 Obj 3 Solve Systems of Linear Equations by
Graphing.

• Try p 283 36. (first without a calculator, then
  with!)
• Directions Solve each system by graphing. If
  there is no solution or an infinite number of
  solutions, so state. 2x - y 0
• y 2x Return to procedure slide.
• (For more practice try p 281-282 Example 5 and
  check point 5)

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Page 28344 a, b
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5.1 Obj 2 Solve Systems of Linear Equations
by Graphing.

• P284 Writing in Mathematics 47-50
• (also assigned for hw)
• Critical Thinking Exercises 53, and why are the
  others false?
• Try 54-56 (not assigned for hw)
• 57 explains how to use the graphing calculator
  to check your graphical solutions.

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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.