Simplification 

Algebra contains numbers as well as alphabetic symbols. When this algebraic expression is simplified, the resultant expression is found which is simpler and shorter than the original one. However, there is no standard procedure to do this because there are a lot of different types of expressions. But these different types of expressions can be generalized in 3 kinds.  

The one that can be simplified and thus do not require any preparation 

The one that needs preparation before it is simplified. 

The one which cannot be simplified. 

To simplify any expression, we group the like terms to solve them and rewrite the simplified expression. 

Example: 

5x + 8y -2 +3x + 3y+ 9 

Solution 

This expression can be simplified by identifying like terms and then group these like terms. 

Here, +5X and +3X are like terms. These can be combined to form 8X. 

  +8Y ad +3Y can be combined to form 11Y. 

-2 and +9 are again like terms which combine to form 7. 

So, expression can be simplified to 

8X+11Y+7 

Example: 

5m + (10m -2m +2) + 6 

Solution: 

In this example, the brackets can be removed first and then like previous example like terms can be combined to simplify the expression. 

The brackets can be solved first (10m -2m +2) =(8m +2) 

Remove the bracket: 5m +8m +2 +6 

Grouping the like terms: 13m +8 

So simplified expression is 

13m + 8 

Example: 

3pq - 4az 

Solution 

This expression cannot be simplified further because it does not have any like term which can be combined or grouped together. 

Rules from geometry 

The basic formulae of rectangle and square are already done in measurement chapter. In the following section we are going to rewrite them using algebra. 

Perimeter of square or for that matter any geometrical figure is sum of all its sides. Here, in above figure, length of each side is L and number of sides are 4. So, Perimeter of square is 4 times L i.e. 4L. So, this rule defines the relationship between perimeter and the length of square. In 4L, L is variable and 4 is coefficient. So, it is an algebraic term. 

Let us consider a rectangle, which has four sides. Here opposite sides are equal in length, its two sides are denoted by “L” and the other two sides are denoted by “B”. 

Perimeter of rectangle = Length of side 1 +Length of side 2 +Length of side 3+ Length of side 4 

As we already know opposite side are equal. 

So, 

Perimeter of rectangle =L +B +L +B 

So, the expression can be simplified as = 2L+ 2B 

Where, L and B are the length and breadth of rectangle respectively. 

Note: Here L and B are both independent variables i.e. the value which one variable takes does not impact the value of other variable. 

So, from above examples we can say that concept of variables, expressions, coefficients and constants can be used easily to generalize other geometrical formulas. These include areas and perimeters of any plane figure, volumes and surface dimensions of 3D figures etc.  

Rules from Arithmetic: 

Commutative property of addition 

  We know, 

7+ 8 =15 

And 

8+7 =15 

So, from above we can see that 

7+8 =8+7 

This property is valid for any two whole numbers. This is called commutativity of addition of two numbers. The word “Commuting” is defined as interchanging. Interchanging the order of two numbers does not change the sum. Variables help us in generalizing this expression. 

Let x and y are two variables that can take any value. 

Then, x + y =y +x 

Commutativity of Multiplication of two numbers 

From the chapter of whole numbers, we can see that the multiplication of two numbers remains same irrespective of their order in the expression. 

For example, 

4 × 3 = 3× 4 = 12 

So,      4 × 3 =3× 4 

This property is called commutativity of multiplication of numbers. So it can be generalized as  

a × b =b ×a 

A and B are variables. They can take any values. 

Distributivity of numbers 

Suppose, we are calculating  

7 × (6+24) =7×6 + 7×24 

This is true for any three numbers we are calculating. This property is known as distributivity of multiplication of numbers over addition of numbers. Let x, y and z are three variables which can take any values. 

X × (Y +Z) =X × Y + X×Z 

We can generalize many such properties of numbers using algebra. 

Expression with Variables:  

We have previously covered the definition of expressions. Let’s revise it once before discussing the next concept. 

13 subtracted from Z. Form its expression 

Z- 13 

S multiplied by 8. Form the expression 

8S 

P multiplied by 2 and 5 subtracted from the product. 

2P -5 

Practical Situations: 

There are many practical situations in which expressions are important. 

Example: 

Situation 1: 

Shakeena is 5 times older than Neha 

Solution 

Let Neha’s age is X 

Shakeena’s age is 5X 

Situation 2: 

Aditya has 5 more pens than Janvi 

Solution: 

Let number of pens with Janvi =X 

Number of pens with Aditya =X+5 

Situation 3: 

How old was Raghav 5 years ago? 

Solution: 

Let Raghav’s present Age =x 

Raghav’s age 5 years ago =x -5 

Introduction to Equations: 

Expression can be simply written as  

Y+ 9 

The expression can be equal to any number of variable Y. For example, for Y =3, the value of expression is 12.   

An equation tells that two things are equal. It has an “=” sign. 

        Y +9 =10 

It says that what is on the left i.e. (y + 9) is equal to what is on the right (10). 

Equation acts as a condition on variable. The equation can only be satisfied by particular value/values of variables. For example, above Y +9 =10, is only possible when Y=1. 

Thus, Left Hand Side is (1+9)  

And Right Hand Side is10 

So, equation says that Left Hand Side is equal to Right Hand Side. 

(1+9) =10 

Y+9 =10, is true only when Y=1. 

Another example can be, 2n =18.It can only be satisfied if value of n is 9. So equation 2n = 18 says that left hand side (LHS) is equal to Right hand side (RHS) only for a particular value of variable(n) which in this case is 9. 

Note: 

Any equation has “=” sign in between LHS and RHS. For example, 2x+5 =11 

The statements which have less than (<) or greater than sign (>) in between are not equations. Example (x-6) > 12 

The equation which only has numbers is called numerical equation. Example 2 * 9 =18 

The equation which has variable in it is called algebraic equation. 2p -14 =2 

Solution of an Equation 

The values of the variable in an equation which satisfies right and left hand side of the equations are called solution to an equation. The equation can either be true or false. It depends on the value chosen for variable. So it is necessary to choose the correct value for variable to make equation true.  

For example, 3n =9. For n=3, the equation can become true. So n=3 is the solution of equation 3n=9. 

So, solution of equation is value/values of the variable that helps in making the equation as true statement. 

For example, 

5X + 9 =19 

It is linear equation because maximum power of X is 1 here. The solution to the equation can be calculated by taking constant of LHS to the RHS. 

5X=19-9 

5X=10 

X=2 

Note: When any term moves from one side to another its sign changes. Like above +9 becomes -9 when moved from LHS to RHS. 

We can also form a table (trial and error method) to find solution of equation. 

For example, 

5X +9 =19 

An equation is finding true solution of statement. 

Example: 

Find solution for the following equation. 

6X +9 =39 

Solution 

Step 1: Take all constants on the Right hand side 

6X =39-9 

Step 2: Solve the equation 

6X =30 

X=30/6 

X=5 

Example: 

Find solution for the following equation. 

X ^2 =25 

Solution 

We will solve the question by trial and error method. 

So, solution to equation is 5 i.e. X=5 

Can you solve this? 

2+3=8, 

3+7=27, 

4+5=32, 

5+8=60, 

6+7=72, 

7+8=?? 

Solution: 98 

2 + 3 = 2 × [3 + (2-1)] = 8 

3 + 7 = 3 × [7 + (3-1)] = 27 

4 + 5 = 4 × [5 + (4-1)] = 32 

5 + 8 = 5 × [8 + (5-1)] = 60 

6 + 7 = 6 × [7 + (6-1)] = 72 

therefore 

7 + 8 = 7 × [8 + (7-1)] = 98 

x + y = x [y + (x-1)] = x^2 + xy -x