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MAT091 :: Lecture Note :: Week 12
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Overview
Assignment(s)

Intersecting Lines

Two lines that are not parallel can intersect. If the equations for the lines are known, then the point of intersection can be determined as follows.

   line A:  y = 2x + 2
   line B:  y = 5x + 5

   set the lines equal to one another...
      2x + 2 = 5x + 5

   solve for x...
      2x + 2 + -2 = 5x + 5 + -2    # add -2 to both sides
      2x = 5x + 3

      2x + -5x = 5x + 3 + -5x      # add -5x to both sides
      -3x = 3

      -3x / -3 = 3 / -3            # divide both sides by -3
      x = -1

   substitute for x in one of the line equations and evaluate...
      y = 2(-1) + 2 = -2 + 2 = 0

   point of intersection:  (-1, 0)

   check answer: plug (-1, 0) into the other line equation 
                 and see if it works...
      y = 5(-1) + 5 = -5 + 5 = 0

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Systems of Linear Equations

A "system" of equations is a collection of equations that are treated as an atomic unit. In other words, a system of equations is used when dealing with more than one variable.

   A: 3x + 3y = 0
   B: 12x + 2y = -20

   pick one of the equations...
      pick A

   solve equation A for y
      3x + 3y = 0
      -3x + 3x + 3y = 0 + -3x
      3y = -3x
      3y / 3 = -3x / 3
      y = -x

   substitute for y in the equation B...
      12x + 2(-x) = -20

   solve equation B for x...
      12x + -2x = -20
      10x = -20
      10x / 10 = -20 / 10
      x = -2

   pick one of the equations...
      pick A

   substitute for x in the equation A...
      3(-2) + 3y = 0

   solve equation A for y...
      3(-2) + 3y = 0
      -6 + 3y = 0
      6 + -6 + 3y = 0 + 6
      3y = 6
      3y / 3 =  6 / 3
      y = 2

   the solution is the point (-2, 2)

Another system of linear equations.

   Two numbers have a sum of 55 and a difference of 9. 
   Find the numbers.

   let x and y be the two unknown numbers

   form two equations...
      A: x + y = 55
      B: x - y = 9

   add the two equations together...
         x + y = 55
      +  x - y =  9
      =============
        2x = 64

   solve for x...
      2x = 64
      2x / 2 = 64 / 2
      x = 32

   pick an equation, substitute for x, solve for y
      pick A
      32 + y = 55
      -32 + 32 + y = 55 + -32
      y = 23

   solution:  (32, 23)

   [exercise] Re-do this problem using substitution.

Here is another system of linear equations that is solved using the addition method.

   A: 5x + y = 10
   B:  x + y = 20

   multiply both sides of B by -1...
      -1(x + y) = 20(-1)
      -x - y = -20

   add the two equations...
         5x + y =  10
      + -1x - y = -20
      ===============
         4x     = -10

   solve for x...
      4x / 4 = -10 / 4
       x     = -2.5

   pick an equation and solve for y...
      pick B
      -2.5 + y = 20
      y = 22.5

   solution:  (-2.5, 22.5)

   subtitute the solution into A and see if it works...
      5(-2.5) + 22.5 = 10
      -12.5 + 22.5 = 10
      it works!

Is a system of equations necessary to solve the following problem?

   Ten less than half of a number is thirty. What is the number?

No, because there is only one variable.

   n/2 - 10 = 30
   n/2 - 10 + 10 = 30 + 10
   n/2 = 40
   n/2 * 2 = 40 * 2
   n = 80

Here is a exercise that can be solved using a system of equations.

   A new nightclub had a grand opening in Scottsdale.  First night
   admission was $4 for females and $6 for males.  90 people attended
   the grand opening and the nightclub collected $460.  How many 
   females and how many males attended the grand opening?
   let 'f' represent # of females and let 'm' represent # of males

   A: f + m = 90
   B: $4(f) + $6(m) = $460

   pick A and solve for 'f'...
      f = 90 - m

   substitute 'f' into B...
      4(90 - m) + 6m = 460

   distribute the 4, combine like-terms, solve for 'm'...
      360 - 4m + 6m = 460
      360 + 2m = 460
      -360 + 360 + 2m = 460 - 360
      2m = 100
      (1/2)2m = 100(1/2)
      m = 50

   substitute m = 50 into A and solve for 'f'...
      f + 50 = 90
      f = 40

   There were 40 females and 50 males the grand opening.

Education.Yahoo.com:: Solving Linear Systems

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Solving System of Linear Equations Using TI-83

The "intersection" feature of the TI-83 calculator can be used to find the solution to a system of linear equations.

   press Y=
   Y1 press CLEAR, if necessary
   enter equation:  X is entered using key labeled X,T,theta,n
   -2X + 3 ENTER
   cursor at Y2, press CLEAR if necessary
   enter equation:  
   (1/3)X - 4 ENTER
   press GRAPH
   press 2ND TRACE 5 (intersect)
   First curve? ENTER
   Second curve? ENTER
   Guess? ENTER
   Intersection x = 3 and y = -3   (3, -3)

Substitute x = 3 into both equations and test if (3, -3) is a solution.

   f(x) = -2x + 3
   f(3) = -2(3) + 3 = -6 + 3 = -3     check

   f(x) = (1/3)x - 4
   f(3) = (1/3)(3) - 4 = 1 - 4 = -3   check

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