S2T2 Solving Linear and Fractional Equations

 Solving Linear Equation                                      

Equation : 

(1) a mathematical expression containing the symbol "=" 

           L.H.S = R.H.S

(2) often contains ALGEBRA => variables, such as x and y (to represent the number that we do not know)

(3) examples: y = 3, y + 3 = 10, y = 2x + 3

(4) solving an equation => finding the value of the variable/s (unknown).

(5) must follow the order of operations - BIDMAS ( Bracket, Indices, Divide, Multiply, Add, Subtract).

Example

 Solve for x                                                                            

               5x + 7 = 2x + 28

Step 1 : "move" the variable to LHS, and numbers to RHS

                 LHS   RHS

              5x + 7 = 2x + 28

  5x -2x + 7 - 7 = 2x + 28 - 2x - 7

              5x -2x + 7 - 7 = 2x + 28 - 2x - 7

[ when "moving", the equation must remain 'balance", thus -2x and -7 are applied to both side]

Step2 : Do x/+- to solve

              5x – 2x = 28 – 7 

                      3x = 21

                        x = 7

["Move" also => "changing" the sign of the number/variable at the other side (LHS/RHS).]

Example

Solve for 2y + 4 = y -6

      2y + 4 -y - 4 = y - 6 - y - 4 (step 1: move y to LHS, nos to RHS)

     2y + 4 -y - 4 = y - 6 - y - 4

         y = -10

Solving Fractional Equations (Linear equation)                                  

Linear fractional Equation can usually be solved by simplifying the equation to having a single denominator.

Example

  Solve y/2 + y/3 = 2

Step 1 : Find the lowest common factor (LCM) for the denominators

LCM of 2 and 3 = 6

Step 2 : Combine into single fraction

             ½y + ⅓y = 2

             3 x y + 2 x y  = 2

             3 x 2   2 x 3  

     3y + 2y = 2

                  6

  6 x 5y = 2 x 6 (multiply 6 => "1/6(LHS) = x 6 (RHS))

                  6

  5y = 12

  y = 12/5 = 2 2/5

Example

   Solve y  +  y – 2 = 3

             3        4

 LCM of 3 and 4 = 12

      4x y  +  3x (y – 2)  = 3       (Step 1 :Find LCM and multiply)

      4x 3      3x    4

       

      4y + 3y – 6 = 3       (Step2 : Combine into single fraction)

             12

       7x – 6 = 3        (Step3: Cross-over the denominator and solve)

          12

      7x – 6 = 3 x 12

      7x = 36 + 6

       x = 42/7 = 6

Example

Solve    3   = 3

          x – 2 

          3 = 3(x – 2)    (Step1: Bring the denominator to the right)

            3 = 3x – 6     (Step2: Expand the bracket)

            3x – 6 = 3    (Step3: Solve for x)

            3x = 3 + 6

              x = 9/3

                 = 3