MATH 048 by qingyunliuliu

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									MATH 048
SIMULTANEOUS
  EQUATIONS
 Solving Systems of Equations Algebraically-
             Elimination Method
• Solving a system of linear equations
  algebraically using ELIMINATION with addition
  and subtraction is easiest when the equations
  are in standard form(i.e. x and y on both sides
  and constant on other side).
Solving a system of equations by elimination using
addition and subtraction.
 Step 1: Put the equations in            Standard Form: Ax + By = C
    Standard Form.
                                          Look for variables that have the
 Step 2: Determine which            same coefficient. If none then multiply one or
    variable to eliminate.             both equations by an appropriate number
                                               to make same or opposites .

 Step 3: Add or subtract the
                                            Solve for the variable.
    equations.

 Step 4: Plug back in to find the   Substitute the value of the variable
    other variable.                         into the equation.

                                     Substitute your ordered pair into
 Step 5: Check your solution.
                                             BOTH equations.
.        1) Solve the system using elimination

                                   x+y=5
                                   3x – y = 7
    Step 1: Put the equations in                  They already are!
       Standard Form.

                                                The y’s have the same
    Step 2: Determine which                          coefficient.
       variable to eliminate.
                                                 Add to eliminate y.
                                                        x+ y=5
    Step 3: Add or subtract the
                                                   (+) 3x – y = 7
       equations.
                                                       4x = 12
                                                            x=3
  1) Solve the system using                       elimination
                    contd.
                                    x+y=5
                                   3x – y = 7
                                                  x+y=5
Step 4: Plug back in to find the
   other variable.                               (3) + y = 5
                                                    y=2
                                                    (3, 2)
Step 5: Check your solution.                    (3) + (2) = 5
                                                3(3) - (2) = 7
The solution is (3, 2). What do you think the answer would
           be if you solved using substitution?
   Examples. Solve using Elimination

a) x  y  9      b) x  2 y  8            c) x  2 y  9
   x- y 3           x-2y  4                 2 x  18  4 y


d ) m  2n  14   e) 4 y  3x  -1          f ) 9 x  4 y  -25
   3n  m  18       6 y - 7 x  -36            6 x -14 y  -25


g) 2  3  4
   x   y
                  h) 0.04 x  0.05 y  44   i) 2( x  y)  y  1
   x
   4   3 4
        y
                     x  y  1000              3( x  1)  y -3
    Solving Systems of Equations
• These notes show how to solve a system
  algebraically using SUBSTITUTION.

• The method is easiest to use when the system
  is a system of linear equations where one of
  the variable’s coefficient is 1.
     Solving a system of equations by substitution
                                   Pick the easier equation. For mixed
Step 1: Solve an equation for           solving the linear equation for
   one variable.                           variable is usually easier.

                                    Put the equation solved in Step 1
Step 2: Substitute
                                         into the other equation.


Step 3: Solve the equation.             Get the variable by itself.


Step 4: Plug back in to find the   Substitute the value of the variable
   other variable.                         into the equation.

                                    Substitute your ordered pair into
Step 5: Check your solution.
                                            BOTH equations.
    1) Solve the system using substitution

                                x+y=5
                                y=3+x
Step 1: Solve an equation for       The second equation is
   one variable.                     already solved for y!
                                           x+y=5
Step 2: Substitute
                                          x + (3 + x) = 5
                                          2x + 3 = 5
Step 3: Solve the equation.                 2x = 2
                                             x=1
    1) Solve the system using substitution

                                   x+y=5
                                   y=3+x
                                             x+y=5
Step 4: Plug back in to find the
   other variable.                          (1) + y = 5
                                               y=4
                                               (1, 4)
Step 5: Check your solution.               (1) + (4) = 5
                                           (4) = 3 + (1)
The solution is (1, 4). What do you think the answer would
          be if you graphed the two equations?
  Examples. Solve using substitution method

a) y  3x         b) x  4 y       c) a  b  3
  x  y  83        x  8 y  52      a  3b  1


d ) 4x  y  0
   8x  1 y  7
        4
                     More Examples
• Solve the following using any method.

      a ) 3 p  2r  7    b) y  x
          p  2r  11      3 y  3x  4
     Examples - Applications of Systems
• 1) The sum of two numbers is 56. Twice the smaller number exceeds the
  larger number by 22. Find the numbers.

• 2)The sum of two numbers is 50. If twice the larger is subtracted from 4
  times the smaller, the result is 8. Find the numbers.

• 3) Mrs. B bought 3 cans of corn and 5 cans of tomatoes for $1.82. The
  following week, she bought2 cans of corn and 3 cans of tomatoes for
  $1.11,paying the same prices. Find the cost of a can of corn and the cost of
  a can of tomatoes.

								
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