S2T3 Solving Simultaneous Equations

Usually given 2 equations with 2 variables, and to find the values of the variables.

Example

Solve for x and y for the following equations:

      3x + 2y = 21  ------ (1) Equation 1

      4x + y = 8 ---------- (2) Equation 2

The variables are x and y

There are usually three methods to use for Simultaneous Equation:

(1) By Substitution

(2) By Elimination

(3) Graphical

(1) By substitution (when there is a X or Y):

Example

Solve for x and y:      

      3x + 2y = 21

        4x + y = 8

Step 1: look for variable with unit digit (1), eg: x or y

Step 2: Arrange equation as such: 

      4x + y = 8

      y = 8 – 4x

Step 3: Substitute y in the other equation (1) and solve

      3x + 2y = 21

      3x + 2(8 – 4x) = 21

      3x + 16 – 8x = 21

        -5x = 21 – 16

       -5x = 5

           x = -1

Substitute x = -1 into y = 8 – 4x, 

          y = 8 -4(-1)

             = 8 + 4 = 12

          y = 12

[[ Test if values of x and y are correct   ]]

Since we use equation 1 to find y, substitute value of x and y into (2).

3 (-1) + 2(12) = -3 + 24 = 21 => value of x and y are correct!

(2) By Elimination

-  'Getting ride' of 1 variable by 'ADDING' or 'SUBTRACTING' both equations

-   1 of the variables must have the same numerical in both equations.

Example

By ADDING

     5y + 2x = 2  --------- (1)

    -5y - 3x = 8  ---------  (2)

  

Using (1) + (2) 

    5y + 2x = 2

+ -5y - 4x = 8.    [5y + (-5y) = 5y - 5y = 0 ]

     0  -x2 = -6  (eliminated Y , find both x and then y)

          x = -6/-2 = 3

Substitute x = 3 into (1) [to find y]

                5y +2(3) = 2

                  y = 2 - 6 = 4

                  y = 4/5

BY SUBTRACTING

       -3x + 2y = 1  ------- (1)

       -3x + 5x = 7 -------- (2)

Using (1) - (2)

       -3x + 2y = 1  

   -   -3x + 5y = 7      [-3x - (-3x) = -3x + 3x]

               -3y = -6 (eliminated X , find y and then x)

                 y = -6/-3 = 2

    Substitute y = 2 into (1)

             -3x + 2(2) = 1

                 -3x = 1 - 4

                    x = -3/-3

                       = 1

Example

Solve the equations by elimination method.

      3x + 2y = 9 --------- (1)  

      4x + 3y = 8 ----------(2)  

Step1: look for variables with smaller digits, (y:2y and 3y, x: 3x, 4x; choose y)

Step2: Make y in both equations to be the same value => 6y by multiplying

      3x + 2y = 9     x3 

      9x + 6y = 27  -------- (3)

          

      4x + 3y = 8    x2  

      8x + 6y = 16   -------- (4)

Step3: Eliminate by + or –

      [both are 6y, 6y => (3) – (4)  because 6y – 6y = 0] 

(3) – (4)

      9x + 6y – (8x + 6y) = 27 – 16

      9x + 6y – 8x – 6y = 11

                  x = 11

OR place the equation in this way:-

        9x + 6y = 27

  -     8x +6y  = 16

          x + 0  = 11   [ x = 9x – 8x, y is eliminated from 6y – 6y= 0, 11 = 27 – 16]

          x = 11

Sub x = 11 into 1

      3(11) + 2y = 9

      33 + 2y = 9

      2y = 9-33

      2y = 24

        y = 12

By Graphical Method

Value of x and y of 2 equations is at the point of interception of the line of equations on the graph 

2 linear equations:y = 2x – 2

                              y = -2x + 4

From the graph, the intersection coordinate is (3/2, 1) => x = 3/2 and y = 1