Class 7 — Mathematics (NCERT)

Algebraic Expressions

Class 7

✓By the end of this lesson students will be able to identify variables, constants, and terms in an algebraic expression.
✓By the end of this lesson students will be able to distinguish between like and unlike terms.
✓By the end of this lesson students will be able to add algebraic expressions by combining like terms.
✓By the end of this lesson students will be able to subtract algebraic expressions.
✓By the end of this lesson students will be able to simplify algebraic expressions.

Key concepts

Variables

A symbol, usually a letter like x, y, z, a, b, etc., that can take various numerical values. Its value is not fixed and can change.

Constants

A symbol that has a fixed numerical value. Its value does not change. For example, 5, -7, 1/2 are constants.

Terms

The parts of an algebraic expression that are separated by addition (+) or subtraction (-) signs. A term is a product of variables and constants. For example, in 3x + 5, '3x' and '5' are terms.

Algebraic Expression

A combination of variables, constants, and mathematical operations (+, -, ×, ÷). Terms are combined using addition or subtraction. For example, 2x + 3y - 7 is an algebraic expression.

Factors of a Term

Each term in an expression is a product of its factors. These can be numerical (like 5 in 5x) or literal (variable, like x in 5x) factors.

Coefficients

In a term, any factor of the term is called the coefficient of the product of the other factors. Usually, the numerical factor is called the numerical coefficient, and the variable part is called the literal coefficient. For example, in 5xy, 5 is the numerical coefficient, and xy is the literal coefficient.

Like Terms

Terms that have the same literal factors (same variables raised to the same powers). Only their numerical coefficients can be different. For example, 3x and -7x are like terms; 5x²y and x²y are like terms.

Unlike Terms

Terms that have different literal factors (different variables or same variables raised to different powers). For example, 2x and 3y are unlike terms; 4x² and 4x are unlike terms.

Addition of Algebraic Expressions

To add algebraic expressions, we combine the like terms. This can be done by arranging terms in rows (column method) or by collecting like terms horizontally.

Subtraction of Algebraic Expressions

To subtract an algebraic expression from another, we change the sign of each term of the expression to be subtracted (the subtrahend) and then add the resulting expression to the other expression (the minuend).

Key facts to remember

1An algebraic expression is a combination of variables, constants, and arithmetic operations.
2Terms are the individual parts of an expression separated by '+' or '-' signs.
3Like terms have identical literal factors (same variables with the same powers).
4Unlike terms have different literal factors.
5Only like terms can be added or subtracted.
6The numerical factor of a term is called its numerical coefficient.
7When subtracting an expression, change the sign of each term of the expression being subtracted and then add.
8A constant term is a term without any variable part.

Worked examples

Example 1

Identify the terms, their numerical coefficients, and group the like terms in the expression: 5x²y - 3xy + 7x²y - 2xy + 8

IFirst, identify all the terms in the given expression: 5x²y, -3xy, 7x²y, -2xy, 8.

IINext, identify the numerical coefficient for each term:

III - For 5x²y, the numerical coefficient is 5.

IV - For -3xy, the numerical coefficient is -3.

V - For 7x²y, the numerical coefficient is 7.

VI - For -2xy, the numerical coefficient is -2.

VII - For 8, the numerical coefficient is 8 (it is a constant term).

VIIIFinally, group the like terms. Like terms have the same literal factors (same variables with the same powers):

9 - Terms with literal factor x²y: (5x²y, 7x²y)

10 - Terms with literal factor xy: (-3xy, -2xy)

11 - Constant term: (8)

Answer

Terms: 5x²y, -3xy, 7x²y, -2xy, 8\nNumerical Coefficients: 5, -3, 7, -2, 8\nLike Terms: (5x²y, 7x²y), (-3xy, -2xy)

Remember that a constant term like '8' is also considered a term, and its numerical coefficient is itself.

Example 2

Add the expressions: (3x + 5y - 4z) and (2x - 2y + 7z)

IWrite the expressions for addition: (3x + 5y - 4z) + (2x - 2y + 7z)

IIRemove the parentheses: 3x + 5y - 4z + 2x - 2y + 7z

IIIGroup the like terms together: (3x + 2x) + (5y - 2y) + (-4z + 7z)

IVCombine the numerical coefficients of the like terms:

V (3 + 2)x + (5 - 2)y + (-4 + 7)z

VISimplify the expression: 5x + 3y + 3z

Answer

5x + 3y + 3z

You can also use the column method by writing like terms one below the other and then adding them.

Example 3

Subtract (4a - 7b + 3c - 2) from (3a - 8b + 5c + 6)

IWrite the expression for subtraction, ensuring the 'from' expression comes first: (3a - 8b + 5c + 6) - (4a - 7b + 3c - 2)

IITo subtract, change the sign of each term in the expression being subtracted (the subtrahend) and then add.

IIIThe subtrahend is (4a - 7b + 3c - 2). Changing its signs gives: -4a + 7b - 3c + 2.

IVNow, add this to the first expression: (3a - 8b + 5c + 6) + (-4a + 7b - 3c + 2)

VRemove parentheses and group the like terms:

VI 3a - 8b + 5c + 6 - 4a + 7b - 3c + 2

VII (3a - 4a) + (-8b + 7b) + (5c - 3c) + (6 + 2)

VIIICombine the like terms:

9 (3 - 4)a + (-8 + 7)b + (5 - 3)c + (6 + 2)

10Simplify the expression: -1a - 1b + 2c + 8

11Write the final answer: -a - b + 2c + 8

Answer

-a - b + 2c + 8

Be extremely careful with changing signs during subtraction. A common mistake is to change only the sign of the first term in the subtrahend.

Common mistakes

✗Attempting to add or subtract unlike terms (e.g., 2x + 3y = 5xy, which is incorrect).
✗Making sign errors during subtraction, especially by not changing the sign of all terms in the expression being subtracted.
✗Confusing x and x² as like terms; they are unlike terms because the powers of the variable are different.
✗Incorrectly combining coefficients, for example, writing x + x as x² instead of 2x.
✗Forgetting that a variable written alone (e.g., 'y') has a numerical coefficient of 1 (i.e., 1y).

Exam tips

★Always group like terms together before performing addition or subtraction to avoid errors.
★When subtracting expressions, rewrite the subtraction as an addition by changing the sign of every term in the expression being subtracted. This helps prevent sign errors.
★Pay close attention to positive and negative signs, especially when combining terms with different signs.
★Present your steps clearly, showing the grouping of like terms. This not only helps in avoiding mistakes but also earns partial marks in case of a final error.

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