Simultaneous Equation~

     A.   Simultaneous Equation

      1.     Linear equation are equations that have one or more unknowns in the first degree.

      For example, 2y-3x=5

      2.     Non-linear equations have one or more unknowns in degrees greater than one.

      For example: x²+2x+y²-3y=0

      3.     Simultaneous equations consists of two equations that have common solutions, a linear equation and a non-linear equation

      4.     The steps in solving simultaneous equations are:

      Step 1

      From the linear equation, an unknown is expressed in term of the other unknown

      Step 2

      The unknown is substituted into the non-linear equation and a quadratic equation in term of the other unknown is formed 

      Step 3

      Simplify and solve the quadratic equation to obtain the values of  first unknown using factorization or the quadratic formula

      Step 4

      Obtain the values of the second unknown by substituting the values of the first unknown, one by one, into the linear equation.

     Simultaneous linear equations can be solve by using two method which is

      Method 1: Elimination

      Or

      Method 2: Substitution

      Example:  Solve the following simultaneous linear equations:

            x - 3y = 2

            3y + x = 8

     By using method 1:

            x - 3y = 2

          (-) x +3y = 8

                -6y = -6

                   y = 1

      when the value of y is equal to 1

             x - 3(1) = 2

             x – 3=2

             x = 5

             y  = 1, x = 5

      By using method 2:

            x - 3y = 2 ----equation 1

            3y + x = 8 -----equation 2

      From equation 1, express unknown of x in the equation in terms of the other. Then, a new equation is form.

                           x – 3y = 2

                           x = 2 + 3y ------equation 3

      Substitute equation 3 into equation 2

                          3y + (2+ 3y) =8

                          6y = 8-2

                          6y = 6

                            y = 1

     when the value of y is equal to 1,

     x= 2 + 3 (1)

       = 5

     x= 5, y = 1

    Simultaneous equations in two unknowns (one linear equation and one non-linear equation)

      x- y = -2

      x² – 6y² = -17

      x  – y =-2

      x  = -2 +y                       (Express one unknown in the equation in terms of the other)

(-2 +y)² -6y² = -17          (Substitute unknow in equation 1 into non- linear equation to obtain quadratic equation in one unknown)

      (-2 + y)( -2+y) -6y = -17 (Solve the quadratic equation)

      4– 2y – 2y +y² – 6y +17 =0

      y² – 10y +21 =0

     (y – 7)(y – 3) =0

     y= 7, y = 3                                                          

x    = -2+ 3                                             ,           x = - 2 +7

      = 1                                                                   = 5

  Solving Daily Problem by Simultaneous Equation

    Steps:

   1.     Identify the two unknown involved in the situation and represent the two unknown with        suitable symbols.

   2.     Form 2 equations based on the information given in the situation

   3.     Solve the quadratic equation.

   Example:

   1.     Given that the perimeter of a rectangle is 26cm and its area is 40cm ². Find the length and width of  the rectangle. (where length is greater than width)

Let x be the length of rectangle

Let y be the width of rectangle

x + x +y + y= 26

2x + 2y =26

(÷2) , x +y = 13 ------equation 1

          xy = 40 -------- equation 2

From 1, x = 13- y ------  equation 3

Substitute unknown of x into equation 2

(13 – y)( y) = 40

13y – y² – 40 = 0

                  -y² +13 y -40 =0

                  (- y+ 8)( y- 5) =0

                  -y +8=0 , y- 5=0

                  -y =-8  , y=5
                   y = 8   , y=5