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1. Introduction

This chapter deals with algebraic expressions: combining numbers and letters (variables) to represent mathematical relationships.

• Letters represent unknown or variable quantities.
• Numbers are constants.
• Algebraic expressions help us generalize patterns and solve problems systematically.

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2. What is an Algebraic Expression?

• An algebraic expression consists of terms, which can be:
  1. Constants – numbers (like 5, –3)
  2. Variables – letters (like x, y)
  3. Coefficient – numerical factor of a variable (like 7 in 7x)

Example:$$7x + 5y – 3$$7x+5y–3

• Terms: 7x, 5y, –3
• Coefficients: 7 (for x), 5 (for y)
• Constant: –3

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3. Like Terms and Unlike Terms

• Like terms: Terms with the same variables raised to the same powers.
  • Example: 5x² and –2x²
• Unlike terms: Terms with different variables or powers.
  • Example: 3x and 3y, or 2x² and x

Key Rule: Only like terms can be combined.

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4. Addition and Subtraction of Algebraic Expressions

Step 1: Identify like terms

Step 2: Add or subtract their coefficients

Example 1:$$3x + 5x = 8x$$3x+5x=8x

Example 2:$$7x + 3y – 2x + 5y = (7x–2x) + (3y+5y) = 5x + 8y$$7x+3y–2x+5y=(7x–2x)+(3y+5y)=5x+8y

Example 3 (Subtraction):$$(5x + 3y) – (2x – y) = 5x + 3y – 2x + y = 3x + 4y$$(5x+3y)–(2x–y)=5x+3y–2x+y=3x+4y

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5. Multiplication of Algebraic Expressions

Rule: Multiply coefficients and variables separately. For powers, add the exponents if the base is the same.

• Example 1:

$$2x \cdot 3x^2 = 6x^3$$2x⋅3×2=6×3

• Example 2 (with constants):

$$5 \cdot (x + 2) = 5x + 10$$5⋅(x+2)=5x+10

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6. Division of Algebraic Expressions

Rule: Divide coefficients and subtract powers of the same variable.

• Example 1:

$$\frac{6x^3}{2x} = 3x^2$$2x6x3​=3×2

• Example 2:

$$\frac{10x^2y}{5x} = 2xy$$5x10x2y​=2xy

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7. Factorization

Factorization is writing an expression as a product of its factors.

• Step 1: Find common factor in all terms
• Step 2: Take the common factor out

Example 1:$$6x + 9 = 3(2x + 3)$$6x+9=3(2x+3)

Example 2:$$4xy + 8x = 4x(y + 2)$$4xy+8x=4x(y+2)

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8. Using Expressions in Problem Solving

Algebraic expressions can model real-life situations:

• Example 1: Cost of x pens at ₹5 each:

$$\text{Cost} = 5x$$Cost=5x

• Example 2: Total cost of x pencils at ₹2 each and y erasers at ₹3 each:

$$\text{Total Cost} = 2x + 3y$$Total Cost=2x+3y

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9. Evaluation of Expressions

To evaluate an expression:

• Step 1: Substitute the given values for variables
• Step 2: Apply arithmetic operations

Example:$$x^2 + 2x + 3, \quad x = 2$$x2+2x+3,x=2 $$2^2 + 2\cdot2 + 3 = 4 + 4 + 3 = 11$$22+2⋅2+3=4+4+3=11

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10. Summary Table

Concept	Key Points
Term	A number, variable, or product of both
Coefficient	Numerical factor of a term
Constant	Term without variable
Like Terms	Same variable(s) and powers
Unlike Terms	Different variables or powers
Addition/Subtraction	Combine like terms only
Multiplication	Multiply coefficients, add powers
Division	Divide coefficients, subtract powers
Factorization	Take common factor outside
Evaluation	Substitute values and simplify

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11. Tips for the Chapter

1. Always identify like and unlike terms before combining.
2. Factorization often involves common factors first.
3. Check signs carefully during subtraction.
4. Use small examples to verify expressions before solving bigger problems.

Questions

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Section A: Multiple Choice Questions (MCQs) – 10 Questions

1. In the expression $7x + 5$7x+5, the coefficient of $x$x is:
   a) 5
   b) 7
   c) x
   d) 12
2. Which of the following are like terms?
   a) $3x^2$3×2 and $3x$3x
   b) $5ab$5ab and $–2ab$–2ab
   c) $4x$4x and $4y$4y
   d) $2x^2$2×2 and $2y^2$2y2
3. The sum of $3x + 2y$3x+2y and $5x – y$5x–y is:
   a) $8x + y$8x+y
   b) $15x – 2y$15x–2y
   c) $8xy + y$8xy+y
   d) $8x – 3y$8x–3y
4. The product of $2x$2x and $3x^2$3×2 is:
   a) $5x^3$5×3
   b) $6x^2$6×2
   c) $6x^3$6×3
   d) $5x^4$5×4
5. Which of the following is a constant term?
   a) 7x
   b) 5y
   c) –3
   d) 2xy
6. Divide $12x^3y^2$12x3y2 by $3x^2y$3x2y:
   a) $4x^5y^2$4x5y2
   b) $4xy$4xy
   c) $4x^2y$4x2y
   d) $36x^5y^3$36x5y3
7. Factorize $6x + 9$6x+9:
   a) $3(2x + 3)$3(2x+3)
   b) $6(x + 1.5)$6(x+1.5)
   c) $2(3x + 9)$2(3x+9)
   d) Cannot be factorized
8. Which of the following is an algebraic expression?
   a) $7 + 3$7+3
   b) $x + 5$x+5
   c) $5 = x$5=x
   d) $2 + 3 = 5$2+3=5
9. The difference between $7x + 3$7x+3 and $3x + 8$3x+8 is:
   a) $4x – 5$4x–5
   b) $10x + 11$10x+11
   c) $4x + 11$4x+11
   d) $–4x – 5$–4x–5
10. The sum of $–2ab + 3a^2$–2ab+3a2 and $5ab – a^2$5ab–a2 is:
    a) $3a^2 + 3ab$3a2+3ab
    b) $2a^2 + 3ab$2a2+3ab
    c) $2a^2 – 3ab$2a2–3ab
    d) $–a^2 + 3ab$–a2+3ab

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Section B: Fill in the Blanks – 10 Questions

11. The numerical factor of a term is called its _______.
12. Terms having the same variables with same powers are called _______.
13. The sum of $5x + 3x$5x+3x is _______.
14. The difference of $7y + 4$7y+4 and $3y – 2$3y–2 is _______.
15. Multiply $4x$4x by $3x^2$3×2 = _______.
16. Divide $10x^3y^2$10x3y2 by $2xy$2xy = _______.
17. Factorize $12ab + 18ac$12ab+18ac = _______.
18. The constant term in $5x – 7$5x–7 is _______.
19. Combine like terms: $6x + 3y – 2x + 5y$6x+3y–2x+5y = _______.
20. Expression $2(x + 5)$2(x+5) = _______ when expanded.

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Section C: Short Answer Questions – 10 Questions

21. Write all terms and coefficients in $7x^2 + 3x – 5$7×2+3x–5.
22. Add: $4x – 2y + 3$4x–2y+3 and $–x + 5y – 7$–x+5y–7
23. Subtract: $5a + 2b – 3$5a+2b–3 from $8a – 3b + 4$8a–3b+4
24. Multiply: $2x$2x by $3x + 5$3x+5
25. Multiply: $(x + 3)(x + 4)$(x+3)(x+4)
26. Divide: $6x^3y^2$6x3y2 by $2xy$2xy
27. Factorize: $15x + 25$15x+25
28. Factorize: $12xy – 18x$12xy–18x
29. Evaluate $3x + 5$3x+5 when $x = 2$x=2
30. Evaluate $2x^2 + 3xy – y^2$2×2+3xy–y2 when $x = 1, y = 2$x=1,y=2

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Section D: Long Answer / Problem-Solving – 10 Questions

31. Combine like terms: $7x + 4y – 3x + 2y – 5$7x+4y–3x+2y–5
32. Simplify: $3a + 5b – 2a + 4b + 6$3a+5b–2a+4b+6
33. Multiply: $(2x + 3)(x + 4)$(2x+3)(x+4)
34. Multiply: $(3a – 2b)(2a + 5b)$(3a–2b)(2a+5b)
35. Divide: $10x^3y^2 ÷ 2x^2y$10x3y2÷2x2y
36. Factorize: $8x + 12$8x+12
37. Factorize: $6xy + 9x$6xy+9x
38. Evaluate $x^2 + 3x + 2$x2+3x+2 for $x = 3$x=3
39. Evaluate $2x^2 – xy + y^2$2×2–xy+y2 for $x = 2, y = 1$x=2,y=1
40. A pen costs ₹x, an eraser costs ₹y. Write expression for total cost of 5 pens and 3 erasers.

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Section E: Application / Higher-Order Thinking – 10 Questions

41. Simplify: $4x + 5y – 2x + 3y – 7$4x+5y–2x+3y–7
42. Factorize and simplify: $12a + 18b – 6c$12a+18b–6c
43. Expand and simplify: $3(x + 4) + 2(x – 1)$3(x+4)+2(x–1)
44. Evaluate: $5x + 3y – 2$5x+3y–2 for $x = 2, y = 1$x=2,y=1
45. Write an expression for total marks of x maths papers, y science papers, and z english papers each scored 5, 4, 3 marks respectively.
46. Factorize: $14x + 21y$14x+21y
47. Divide: $15x^4y^2 ÷ 3x^2y$15x4y2÷3x2y
48. Multiply: $(x – 2)(x + 3)$(x–2)(x+3)
49. A rectangle has length $x + 3$x+3 and width $x + 2$x+2. Write expression for area.
50. A shop sells x pens at ₹5 each and y pencils at ₹2 each. Write expression for total cost and evaluate for x = 4, y = 3.

Answers – Chapter 4: Expressions Using Letters and Numbers

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Section A: MCQs – Answers

1. b) 7 – Coefficient of x in $7x + 5$7x+5 is 7.
2. b) 5ab and –2ab – Same variables, same powers → like terms.
3. a) 8x + y – $3x + 2y + 5x – y = 8x + y$3x+2y+5x–y=8x+y
4. c) 6x³ – Multiply coefficients: 2×3=6; add powers of x: 1+2=3 → 6x³
5. c) –3 – Only number without variable is constant.
6. b) 4xy – Divide coefficients: 12 ÷ 3 = 4; subtract powers: x³ ÷ x = x²; y² ÷ y = y → 4x²y? Wait carefully: $12x^3y^2 ÷ 3x^2y = 4x^(3–2)y^(2–1) = 4xy$12x3y2÷3x2y=4x(3–2)y(2–1)=4xy ✔
7. a) 3(2x + 3) – Factor out common 3.
8. b) x + 5 – Expression with letters and numbers.
9. a) 4x – 5 – Subtract: $7x + 3 – (3x + 8) = 7x + 3 – 3x – 8 = 4x – 5$7x+3–(3x+8)=7x+3–3x–8=4x–5
10. a) 3a² + 3ab – Add: (–2ab + 5ab = 3ab), (3a² – a² = 2a²)? Check carefully: $–2ab + 5ab = 3ab$–2ab+5ab=3ab, $3a² – a² = 2a²$3a2–a2=2a2 → Correct answer b) 2a² + 3ab ✔

Correction: Answer is b) 2a² + 3ab

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Section B: Fill in the Blanks – Answers

11. Coefficient
12. Like terms
13. 8x – 5x + 3x = 8x
14. 4y + 6 – (7y + 4) – (3y – 2) = 7y + 4 – 3y + 2 = 4y + 6
15. 12x³ – 4x × 3x² = 12x³
16. 5x²y – 10x³y² ÷ 2xy = 5x²y
17. 6a(b + 3c)? Wait: 12ab + 18ac → common factor 6a → 6a(b + 3c) ✔
18. –7 – constant term in 5x – 7
19. 4x + 8y – 6x – 2x = 4x; 3y + 5y = 8y → 4x + 8y
20. 2x + 10 – 2(x + 5) = 2x + 10

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Section C: Short Answer – Answers

21. Terms: 7x², 3x, –5; Coefficients: 7, 3, –
22. Add: $4x – 2y + 3 + (–x + 5y – 7) = 3x + 3y – 4$4x–2y+3+(–x+5y–7)=3x+3y–4
23. Subtract: $8a – 3b + 4 – (5a + 2b – 3) = 3a – 5b + 7$8a–3b+4–(5a+2b–3)=3a–5b+7
24. Multiply: $2x(3x + 5) = 6x² + 10x$2x(3x+5)=6×2+10x
25. Multiply: $(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12$(x+3)(x+4)=x2+4x+3x+12=x2+7x+12
26. Divide: $6x³y² ÷ 2xy = 3x²y$6x3y2÷2xy=3x2y
27. Factorize: 15x + 25 = 5(3x + 5)
28. Factorize: 12xy – 18x = 6x(2y – 3)
29. Evaluate $3x + 5$3x+5 for x = 2 → 3×2 + 5 = 11
30. Evaluate $2x² + 3xy – y²$2×2+3xy–y2 for x = 1, y = 2 → 2×1 + 3×1×2 – 4 = 2 + 6 – 4 = 4

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Section D: Long Answer / Problem-Solving – Answers

31. Combine like terms: $7x + 4y – 3x + 2y – 5 = 4x + 6y – 5$7x+4y–3x+2y–5=4x+6y–5
32. Simplify: $3a + 5b – 2a + 4b + 6 = a + 9b + 6$3a+5b–2a+4b+6=a+9b+6
33. Multiply: $(2x + 3)(x + 4) = 2x² + 8x + 3x + 12 = 2x² + 11x + 12$(2x+3)(x+4)=2×2+8x+3x+12=2×2+11x+12
34. Multiply: $(3a – 2b)(2a + 5b) = 6a² + 15ab – 4ab – 10b² = 6a² + 11ab – 10b²$(3a–2b)(2a+5b)=6a2+15ab–4ab–10b2=6a2+11ab–10b2
35. Divide: $10x³y² ÷ 2x²y = 5xy$10x3y2÷2x2y=5xy
36. Factorize: 8x + 12 = 4(2x + 3)
37. Factorize: 6xy + 9x = 3x(2y + 3)
38. Evaluate: $x² + 3x + 2$x2+3x+2 for x = 3 → 9 + 9 + 2 = 20
39. Evaluate: $2x² – xy + y²$2×2–xy+y2 for x = 2, y = 1 → 2×4 – 2×1 + 1 = 8 – 2 + 1 = 7
40. Total cost of 5 pens (₹x each) and 3 erasers (₹y each): $5x + 3y$5x+3y

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Section E: Application / Higher-Order Thinking – Answers

41. Simplify: $4x + 5y – 2x + 3y – 7 = 2x + 8y – 7$4x+5y–2x+3y–7=2x+8y–7
42. Factorize: 12a + 18b – 6c → 6(2a + 3b – c)
43. Expand: 3(x + 4) + 2(x – 1) = 3x + 12 + 2x – 2 = 5x + 10
44. Evaluate: 5x + 3y – 2, x = 2, y = 1 → 10 + 3 – 2 = 11
45. Total marks: x maths papers × 5 + y science ×4 + z english ×3 → 5x + 4y + 3z
46. Factorize: 14x + 21y = 7(2x + 3y)
47. Divide: 15x⁴y² ÷ 3x²y = 5x²y
48. Multiply: (x – 2)(x + 3) = x² + 3x – 2x – 6 = x² + x – 6
49. Rectangle area: length × width = (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6
50. Total cost: 5x + 2y; for x = 4, y = 3 → 5×4 + 2×3 = 20 + 6 = 26