Introduction
In everyday life, we often come across situations where a number is unknown. For example,
- The cost of buying notebooks depends on how many notebooks we buy.
- The total fare of a taxi depends on the distance travelled.
- The number of chocolates left depends on how many were eaten.
Instead of writing different calculations every time, we use letters to represent unknown numbers. These letters are called letter-numbers or variables.
An expression formed using numbers, operations, and letter-numbers is called an algebraic expression.
Formation of an expression
Let us look at a few expressions and their statements.
| Statement | Expression |
| \(5\) more than a number/A number is increased by \(5\) | \(x + 5\) |
| \(7\) less than a number | \(x - 7\) |
| Double a number | \(2x\) |
| Triple a number | \(3x\) |
| Half a number | \(\frac{x}{2}\) |
| \(8\) times a number | \(8x\) |
| A number is decreased by \(12\) | \(x - 12\) |
| Two less than \(13\) times a number | \(13x - 2\) |
Real-life situations represented by algebraic expressions
Algebra helps us describe many everyday situations.
| Situation | Expression |
| Cost of \(x\) pens at \(₹15\) each | \(15x\) |
| Fare of \(₹50\) plus \(₹12\) per \(km\) for \(k \ km\) | \(50 + 12k\) |
| Age of a mother \(28\) years older than her son | \(x + 28\) |
| Total marks after scoring \(p\) in Maths and \(q\) in Science | \(p + q\) |
| Money saved after spending \(₹200\) from the salary of \(₹x\) | \(x - 200\) |
Algebraic expressions are combinations of variables, coefficients, and constants separated by operations (\(+, -, \times, \div\)). Operations on algebraic expressions can be performed only on like terms.
Like terms
Terms having the same variables are called like terms.
Example:
- \(3x\) and \(2x\) are like terms. [Having same variable \(x\)]
- \(16ab\) and \(2ab\) are like terms. [Having same variable \(ab\)]
- \(7y + 3 + 5y + 1\). [Here \(7y\) and \(5y\) are like terms because of variable \(y\) and \(3\) and \(1\) are constants]
Unlike terms
Terms having different variables are called unlike terms.
Example:
- \(5x\) and \(8y\) are unlike terms. [Having different variables \(x\) and \(y\)]
- \(3xy\) and \(12y\) are unlike terms [Having different variables \(xy\) and \(y\)]
These cannot be combined.
Simplification of expressions
1. Group like terms together and then add/subtract the like terms.
2. Unlike terms remain separate.
3. Use the distributive property whenever brackets are present. \(a(b + c) = ab + bc\).
Example 1: Add the expression \(2x + 7x\)
Here, \(2x\) and \(7x\) are like terms. To add the expression, let us add its coefficients.
\(2x + 7x = (2 + 7)x\)
\(= 9x\)
Example 2: Subtract the expression \(7xy - 15xy + 3y\)
Here, \(7xy\) and \(15xy\) are like terms and \(3y\) is an unlike term.
\(7xy - 15xy + 3y = (7 - 15)xy + 3y\)
\(= -8xy + 3y\)
Example 3: Simplify the expression \((12ab + 5d) - (4ab - 7d)\)
Let us combine the like terms.
\((12ab + 5d) - (4ab - 7d) = 12ab + 5d - 4ab + 7d\)
\(= (12 - 4)ab + (5 + 7)d\)
\(= 8ab + 12d\)
Therefore, the expression simplifies to \(8ab + 12d\).
Verification of algebraic expression
To verify means to substitute given number values for the letters and calculate, then check whether the simplified and original expressions match or whether the answer matches what's expected.
Verify that \(3(2w + 3x + 4w)\) simplifies correctly.
Simplification:
\(3(2w + 3x + 4w) = 3(6w + 3x)\)
\(= 18w + 9x\)
Thus, the simplified form is \(18w + 9x\).
Verification:
A simplified expression must give the same value as the original expression for any value you substitute.
Substitute \(w = 1\) and \(x = 1\) in both original and simplified expressions.
Original expression: \(3(2w + 3x + 4w) = 3[2(1) + 3(1) + 4(1)]\)
\(= 3(2 + 3 + 4)\)
\(= 3(9)\)
\(= 27\)
Simplified expression: \(18w + 9x = 18(1) + 9(1)\)
\(= 18 + 9\)
\(= 27\)
Thus, \(18w + 9x\) is the correct simplification of the given expression.
Hence, verified.
Applying expressions to real-life problems
Let us understand using an example.
One notebook costs \(₹45\). If \(n\) notebooks are purchased, then find the total cost of the notebooks. Also, simplify the expression when \(n\) is \(8\).
Solution:
Total cost of the notebook \(= 45n\)
Simplification:
When \(n = 8\), then the expression \(45n\) becomes:
Total cost \(= 45n\) \(= 45(8) = ₹360\)