Chapter 11: Algebra

Introduction
The branch of mathematics in which we studied numbers is arithmetic.    
In Arithmetic, numerals 1, 2, 3, 4 , -------- etc. and four fundamental operations : addition , subtraction, multiplication and division are used to deal with various problems. In Algebra, in addition to numerals, we use letters such as x, y, z in various situations to solve the problems.

Matchstick Patterns
Ram and Sarita are making patterns with matchsticks. They decide to make simple patterns of the letters of the English alphabet. Ram takes two matchsticks and forms the letter L as shown in Fig (1). Then Sarita also picks two sticks, forms another letter L and puts it next to the one made by Ram Fig (2). Then Ram adds one more L and this goes on as shown by the dots in Fig (3).

Number of matchsticks required = 2 × number of Ls. For convenience, let us write the letter n for the number of Ls. Thus, n can be any natural number 1, 2,  3, 4, 5, .... We then write, Number of matchsticks required = 2 × n. Instead of writing 2 × n, we write 2n. Note that 2n is same as 2 × n.

Variable
In the above example, we found a rule to give the number of matchsticks required to make a pattern of Ls. The rule was: Number of matchsticks required = 2n Here, n is the number of Ls in the pattern, and n takes values 1, 2, 3, 4,.... 
Let us look at Table 1 once again. In the table, the value of n goes on changing (increasing). As a result, the number of matchsticks required also goes on changing (increasing). 
n is an example of a variable. Its value is not fixed; it can take any value 1, 2, 3, 4, ... . We wrote the rule for the number of matchsticks required using the variable n.
The word ‘variable’ means something that can vary, i.e. change. The value of a variable is not fixed. 
It can take different values.
Examples of variables : We have used the letter n to show a variable. There is nothing special 
about n, any letter can be used. One may use any letter as m, l, p, x, y, z etc. to show a variable. Remember, a variable is a number which does not have a fixed value. For example, the number 5 or the number 100 or any other given number is not a variable. They have fixed values. Similarly, the number of angles of a triangle has a fixed value i.e. 3. It is not a variable. The number of corners of a quadrilateral (4) is fixed; it is also not a variable. But n in the examples we have looked is a variable. It takes on various values 1, 2, 3, 4, ... .    

Let us take other example of variable : Raju and Balu are brothers. Balu is younger than Raju by 3 years. When Raju is 12 years old, Balu is 9 years old. When Raju is 15 years old, Balu is 12 years old. We do not know Raju’s age exactly. It may have any value. Let x denote Raju’s age in years, x is a variable. If Raju’s age in years is x, then Balu’s age in years is (x – 3). The expression (x – 3) is read as x minus three. As you would expect, when x is 12, (x – 3) is 9 and when x is 15, (x – 3) is 12.

Expressions with Variables
To form expressions we use all the four number operations of addition, subtraction, multiplication and division. For example, to form (2 × 10) + 3, we have multiplied 2 by 10 and then added 3 to the product. Expressions can be formed from variables too. In fact, we already have seen expressions with variables, for example: 2n, 5m, x + 10, x – 3 etc. These expressions with variables are obtained by operations of addition, subtraction, multiplication and division on variables. For example, the expression 2n is formed by multiplying the variable n by 2; the expression (x + 10) is formed by adding 10 to the variable x and so on.
One important point must be noted regarding the expressions containing variables. A number expression like (4 × 3) + 5 can be immediately evaluated as (4 × 3) + 5 = 12 + 5 = 17. But an expression like 
(4x + 5), which contains the variable x, cannot be evaluated. Only if x is given some value, an expression like (4x + 5) can be evaluated. For example, when x = 3, 4x + 5 = (4 × 3) + 5 = 17 as found above.

Ex.1     Give expressions for the following cases.
(a)     Rita scores x marks in Maths and 46 marks in English. What is her total score in Maths and English. 
 (b)     The difference of x and 9 where x > 9.
(c)     The product of a and b  added to the difference of a and b (a > b).
(d)     One-half of a multiplied by the sum of x and y.

Ex.2 Ali is x years old. Express the following in algebraic form : 
(a)     4 times Ali's age 3 years hence. 
(b)      The present age of Ali's aunt who is four times as old as Ali will be 5 years from today. 
(c)     The present age of Ali's father who is 5 times as old as Ali was 3 years ago. 

Sol.   (a)     Ali's age 3 years hence = (x + 3) years 4 times Ali's age 3 years hence = 4(x + 3) years. 
(b)      Ali's age after 5 years = (x + 5) years. Ali's aunt age = 4 (x + 5) years   
(c)     Ali's age before 3 years = (x – 3) years. Ali's father age = 5(x – 3) years

Literal Numbers
We have mentioned earlier that the letters represent the numbers. These letters are called literal numbers and obey all the rules of arithmetic. 
Note :  5 × p × q = 5pq. 5, p and q are factors of 5pq. 5 is a numerical factor and p, q are literal factors.

Power of literal numbers 
We have read earlier that 2 x 2 x 2 = 2³ and (–3) x (– 3) = (–3)²      Similarly, a x a x a = a³ and (– y) x (– y) x (– y)  x (– y) = (–y)⁴ 
a³ is read as 'a to the power three' or 'a raised to the power three' or 'a cube' or 'third power of a' and (–y)⁴ is read as '–y to the power four' or '–y raised to the power four' or fourth power of –y'. 
In a³, a is called base and 3 is called exponent or index. 

Coefficient
The number expressed in figures or symbols, standing before an algebraic term as a multiplier is called its coefficient. Thus in 3abc, 3 is the coefficient of abc, 3a is the coefficient of bc and 3ab is the coefficient of c. 

Ex.1     Write down the coefficient of :
(a) x in 3xy    (b) abc in – 5abc     (c) y in 2xyz   (d) a² in – a²bc 

Sol.     (a) 3y             (b) – 5            (c) 2xz             (d) – bc 

Ex.2 Write down the numerical coefficient in each of the following :
           (a) 5 ab         (b) – 3xyz      (c) px          (d) –y 
Sol.    (a) 5              (b) –3            (c) 1            (d) – 1 

Ex.3     If a = 2, b = 3, c = 4, find the value of : 
           (a)     3a – b + 2c 
           (b)     a² – b² + c²        
           (c)     ab – 3abc – 2ac 

Sol.    (a)     3a – b + 2c = 3 x 2 – 3 + 2 x 4 = 6 – 3 + 8 = 11 
          (b)     a² – b² + c²  = 2² – 3² + 4² = 4 – 9 + 16 = 11 
         (c)     ab – 3abc – 2ac = 2 x 3 – 3 x 2 x 3 x 4 – 2 x 2 x 4 = 6 – 72 – 16 = – 82 

Algebric Equation     
An equation is a mathematical statement equating two quantities. An equation has an equal sign (=) between its two sides. The equation says that the value of the left hand side (LHS) is equal to the value of the right hand side (RHS). If the LHS is not equal to the RHS, we do not get an equation.
Equation is satisfied only for a definite value of the variable.
There is an equal sign between the LHS and RHS. Neither of the two sides contain a variable. Both contain numbers. We may call this a numerical equation. Usually, the word equation is used only for equations with one or more variables.
Here are some examples of equations : 2x– 5 = 7, –3x + 2 = 5, y + 3 = 0

Solution of an equation
The value of the variable (unknown) in an equation which satisfies the equation is called a solution to the equation.
The values is called the root(s) of the equation or solution of the equation.
Let us take an example of an equation x + 3 = 10.
We have to find the value of x which will satisfy the above equation. And we observe that if we put 
x = 7 in this equation it will satisfy the equation. So x = 7 is the solution or root of this equation.

Solution of an equation by trial an error
One of the simplest ways of solving an equation is by the trial-and-error method. In this, a guess is made about the value of x, and this value is then substituted in the equation to check if it is the root of the equation. Consider the following example : 4x + 3 = 23
Our equation is 4x + 3 = 23. So we substitute different values for x and try to find out which value of x will satisfy the equation. Make a chart as follow.

Ex.1    Determine if 3 is the root of the equation 5x – 10 = 5.
Sol.     If we put x = 3, then L.H.S. = 5x – 10 = 5 × 3 – 10 = 15 – 10 = 5
        R.H.S. = 5 
        ∴     L.H.S. = R.H.S
        Thus, 3 is a root of the given equation.

Ex.2  If 20 is subtracted from a number, the result is 45. Convert this statement into an algebraic equation.
Sol.    Let us suppose that x is the unknown number.
Then x – 20 stands for 20 subtracted from the number x. This is equal to 45.
Hence, x – 20 = 45 
Once you convert a statement into an algebraic equation, it is easier to solve and find the root.

Ex.3     Solve the equation x – 7 = – 2 and check the result.
Sol.    We have, x – 7 = – 2.
In order to solve this equation, we have to get x by itself on the L.H.S., We need to shift – 7. This can be done by adding 7 to both sides of the given equation. Thus,
        x – 7 = – 2
       ⇒    x – 7 + 7 = – 2 + 7           [Adding 7 to both sides]
       ⇒    x + 0 = 5             [∵ – 7 + 7 = 0 and – 2 + 7 = 5]
       ⇒   x = 5
        Thus, x = 5 is the solution of the given equation.
        L.H.S. = 5 – 7 = – 2 and R.H.S. = – 2
        Thus, when x = 5, we have L.H.S. = R.H.S.

    Ex.4      Solve : 3(x + 3) – 2 (x – 1) = 5 (x – 5).
    Sol.       We have,
     3(x + 3) – 2(x – 1) = 5(x – 5)
    ⇒    3x + 9 – 2x + 2 = 5x – 25    [Expanding brackets on both side]
   ⇒    3x – 2x + 9 + 2 = 5x – 25
  ⇒    x + 11 = 5x – 25        [Taking 5x to the L.H.S. and 11 to the R.H.S.]
  ⇒    – 4x = – 36
   ∴    x = 9