Solving Linear Systems

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					Solving Linear Systems
Using Substitution, Elimination,
and Graphing to solve a system of
equations
Table of Contents
  Review Graphing A Line
  Define a Linear System
  Slope Intercept Form
  Standard Form of a Line
  Solving a Linear System
  Parallel Lines
  Perpendicular Lines
  Choose between Substitution and Elimination
  Solving a System using Substitution
  Solving a System using Elimination
Review of Graphing a Line
  Put equation into slope intercept form
     Practice – Solve for y: 4x - 3y = -9
  Identify the slope and the y-intercept
  Locate the y-intercept on the y-axis
  Starting at that point, move up (+) or
  down(-), then right
     Show me how
                                         Continue
Solve for y: 4x - 3y = -9
  Step 1: Subtract the X term
     - 3y = - 4x – 9
  Step 2: Divide by the coefficient of y
     Do NOT divide by y, JUST the coefficient
     -3y = -4x -9
      -3   -3 -3
  End result:       y = 4x + 3
                        3
                                          Back
Graph a line example
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                       Continue
Graphing Practice
  Graph the following equations:
     5x – 6y = 8

     2x + 4y = -6

     y = -3 x + 2
          2
Define a Linear System
  2 linear equations create a linear system

  Can be in slope intercept or standard
  form
Y form (Slope Intercept Form)
  Slope intercept form y=mx+b

  m is the slope (steepness of line)
  b is the y-intercept (shows where the
  line crosses the y axis)
  Y intercept is also where x = 0 (0,b)
Standard Form of a Line
  Ax + By = C
  A, B, and C are integers
     NO FRACTIONS, NO DECIMALS
     A, B and C can be NEGATIVE
Solving a Linear System
  What does this mean?
     Locating where the lines cross
     Where lines share a coordinate (x,y)


  How many solutions?
     Parallel lines – no solutions
     Same equations – infinite solutions
     Any other lines – one solution
        Perpendicular lines
Parallel Lines
  Same slope
  Lines don’t cross – no solution
  Solve equation – answer is two different
  numbers
  EXAMPLE: y=3x+4
                y=3x-5
     If slope is the same and y-intercept isn’t,
      the lines are parallel
Same Equations
  Place both into slope intercept form
  If the equations are identical
     Lines are located on top of one another
     End result – ONE line
Perpendicular Lines
  One solution – the lines cross
  Cross at a 90 degree angle
  How to tell if two lines are
  perpendicular?
     Put both equations in slope intercept form
     Multiply the slopes
     If answer = -1, they are perpendicular
Choose between Substitution and
Linear Combinations (Elimination)

Solve with Substitution if:
    One equation solves for x (i.e. x=)
    One or both equations are in y form
     (slope-int)
Solve with Linear Combinations
(Elimination) if:
    Both equations are in Standard Form with
     coefficients other than 0
Steps for Solving a System
    with Substitution
Replace (substitute) one variable, and
solve for the other (i.e. replace y and
solve for x, or replace x and solve for y)
Replace that variable in one equation
and solve for the other variable
Check your answers in BOTH equations


                                   Continue
Solving a System with Substitution
              Part 1
 EXAMPLE:
    2x-3y=5
    y=-2x+1


 replace y in first equation with -2x+1
    2x - 3 (-2x + 1) = 5


                                   Continue
Solving a System with Substitution
              Part 2
   Distribute and combine like terms

 2x - 3 (-2x + 1) = 5
 2x + 6x - 3     =5
    8x-3         =5
                Solve for x
               8x = 8
                x=1
                                 Continue
Solving a System with Substitution
              Part 3
       Solve for the second variable
Replace x into one equation
  2(1) - 3 y = 5

Solve for y
     2 - 3y = 5
        - 3y = 3
            y = -1
The solution is (1, -1)
                                   Continue
  Solving a System with Substitution
                Part 4

  Check your answer in both equations
     2x -3y =5
     2(1) - 3(-1) = 5
       2 + 3       =5
                5 =5

y = -2 x +1
  -1 = -2 (1) +1        Both answers check
    -1 = -2        +1                 Continue

  -1 = -1
Solve a System – Elimination
           Part 1

 Be sure both equations are in Standard
 Form
 3x + 5y = 8
-6x – 3y = -2
Solve a System – Elimination
           Part 2

Multiply one or both equations by a
constant
END RESULT – coefficients of one
variable are + and – of same number
2 (3x + 5y = 8)         6x + 10y = 16
  -6x – 3y = -2        -6x – 3y = - 2
Solve an System – Elimination
           Part 3
Combine the equations
 6x + 10y = 16
-6x – 3y = - 2
      7y = 14

Divide by coefficient
   y = 14/7
   y=2
 Solve an System – Elimination
            Part 4
       Solve for the second variable
Replace y into one ORIGINAL equation
  3x + 5 (2) = 8

Solve for y
    3x + 10 = 8
    3x       = -2
          x = -2/3
The solution is (-2/3, 2)
 Solve an System – Elimination
            Part 5
              Check your work
Replace x and y into both equations
   3 x + 5 y =8             -6 x – 3 y = -2
  3(-2/3) + 5 (2) = 8       -6(-2/3)-3(2) = -2
      -2 + 10 = 8                 4 - 6 = -2
            8 = 8                      -2 = -2

Both solutions check   (-2/3, 2)

				
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posted:1/3/2012
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