Chapter 4: Simple equations

Variable
    It means something that can vary or change. It takes different numerical values. Usually variables are denoted by letters of the alphabets. For example, x, y, z, a, b, c etc.

EXPRESSION
    Expressions are obtained by placing mathematical operators like addition, subtraction, multiplication and division on the variables.
    For example, 2x + 3
    This expression is formed by multiplying x by 2 and then adding 3 to the product.

EQUATION
    An equation has always an equality sign. An equation can be formed by using equality sign between two expressions with the condition that these two expressions should have equal value. Atleast one of the two expressions should have equal value. The            equality sign means the value of the expression to the left of the sign is equal to the value of the expression to the right of the sign. 
    An equation remains the same, when the expressions on the left and on the right are interchanged. 
    For example,    2x + 3 = 5x –5 is same as 5x–5 = 2x +3

Properties of linear equations
1.    We can add the same number to both sides of the equation.
    Consider the example x – 8 = 12
    Additing 8 to both sides of the equation, 
    We have x – 8 + 8 = 12 + 8
    ⇒    x + 0 = 20
   ⇒     x  = 20

2.    We can subtract the same number from both sides of the equation.
     Ex.    z + 7 = 8
        Subtracting 7 from both sides of the equation, we have
        z + 7 = 8
        ⇒     z + 7 – 7 = 8 – 7
        ⇒     z + 0 = 1
       ⇒     z = 1

How to solve an equation
1.    Trial and error method

We substitute various values for x and stop only when the value satisfies the equation.

2.    Transposition method: Transposing a number means moving an expression to the other side of the equation
    For example, x +2 = 3
    By transposing 2 from LHS to RHS, we get 
        x = 3 –2    (On transposing +2 become –2)
        x = 1
    Some example of transpostions

Transposition rules:
    (i)    While shifting (+ve), (-ve) terms from LHS to RHS its sign change to (-ve), (+ve) respectively.
    (ii)    While moving the expression from LHS to RHS, which are in multiplication or division expression which are in multiplication moves in division.

Important points

Application of Simple Equation
    Illustration 1
        Five times of a number is 3 more than the two times of a number
    Solution
        Let the unknown number is x
        then five times of number is 5x
        and two times of number is 2x
        and the sum of 2x and 3 is 5x
        ⇒     5x = 2x + 3
        For solving this equation we transpose +2x from RHS to LHS
       ⇒     5x – 2x = 3   ⇒    3x = 3        ⇒    x = 1

Illustration 2
        Construct an equation from the following sentences
        (i)     The sum of 5 and a number is 11
        (ii)     If 7 is subtracted from a number then we get 2
        (iii)    The sum of three times of a number and 5 is equal to 8
                (iv)     One fifth of a number is equal to 2
    Solution
        (i)     Let us say the numbers is x
               Sum of 5 and a number = 5 + x
               So the equation is 5 + x = 11
        (ii)   Let the number is y
               by subtracting 7 from the number we get
                       y –7
              so the equation is y–7 = 2
        (iii)   Let number is z
               three times = 3z
               Sum of 5 and 3z is 3z + 5
               So the equation is 3z + 5 = 8
        (iv)  Let the number is x
              One fifth of number is x/5
              So the equation is x/5 = 2

Illustration 3
        Solve: 3x –2 = 1 and check the result.
    Solution
        By transposing (–2) to RHS
            3x –2 + 2 = 1 + 2    ⇒    3x = 3
        Divide both side by 3
                   
        Check putting value of x = 1 in the LHS of the equation
            LHS = 3 × 1 –2 = 3 –2 = 1 = RHS
            LHS = RHS

Illustration 4
        A’s age is 3 yrs less than two times A’s friend’s age. If A’ age is 5 yrs. Set up an equation to find A’ friend’s age.
    Solution
        Let the A’s friend age is x yrs.
        Two times of x is 2x.
        Age of A is 3 yrs less than 2x
        So the equation is  2x –3 = 5
        Where x is the A’s friend age.

Illustration 5
        One number is 3 less than the two times of the other. If their sum is increased by 7, the     result is 37. Find the numbers.
    Solution
        Let one number = x
            other number = 2x – 3
       ⇒     (x + 2x – 3) + 7 = 37   ⇒     3x + 4 = 37        ⇒     3x = 37 – 4            ⇒     3x = 33        ⇒     x = 11
        ∴    2x – 3 = 2 ´ 11 – 3 = 22 – 3 = 19
        ∴The two numbers are 11 and 19.

Illustration 6
        Construct 2 equations starting with x = 1
    Solution
        (i)     Multiply both sides by 2
                      2x = 2
            Subtract both sides by 1
                2x – 1 = 1