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

                   

Probability Distributions

Probability Distributions

A. Discrete Random Variables

1. A random variable that has finite and countable values is known as a discrete random variable.

2. For example, two coins are tossed simultaneously and the number of tails obtained is studied. If X represents the number of tails obtained, then X can take the values 0, 1 and 2 based on the following table.

Outcomes	TT	TH	HT	HH
X	2	1	1	0

Probability of an Event that follows a Binomial Distribution

1. A trial with only two outcomes i.e. 'success' or 'failure' is known as a Bernoulli trial.

2. If a Bernoulli trial is repeated many times, the experiment is known as the binomial experiment.

3. Let X be a discrete random variable that represents the number of success of a binomial experiment. X follows a binomial distribution (with the number of experiments = n and the probability of success = p) and it is denoted by

Example

Mean, Variance and Standard Deviation of a Binomial Distribution

If X is a binomial discrete random variable such that X~B (n,p), then

• Mean of X = np
• Variance of X = npq
• Standard deviation of X = √ npq

B. Normal Distribution

1. Continuous random variable is a variable that can take any infinite value in a certain range.

2. For example, in a Form 4 class, the mass of the heaviest student is 70 kg and the mass is lightest student is 50 kg. If X represents the masses of any student in that class, then X can take any value from 50 kg to 70 kg such as 60 kg, thus

X = { x: 50 kg ≤ x ≤ 70 kg, x is the mass of students }

3. If a normal random variable, X, has a mean, µ = 0 and a standard deviation σ = 1, then X follows a standard normal distribution, i.e. X~N (0,1).

 4. The variable of a normal distribution can be converted to the variable of the standard normal distribution using the following formula.

                                                                                                                                                
z-Score of a Normal Distribution


  

To Read the Standard Normal Distribution Tables

Probability of an Event that follows a Normal Distribution

To Find Mean and Standard Deviation of a Normal Distribution

Further Examples on Normal Distrbution