Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Introduction to Algebra — Exercise Solutions
How to Use This Page
This page provides step-by-step solutions for every exercise in Class 6 Maths Term 1 Chapter 2: Introduction to Algebra (Samacheer Kalvi / TN State Board). Work through each problem on your own first, then check your working against the solutions here. Every equation is solved with full working shown — transposing terms, simplifying, and verifying the answer.
Key Concepts (Quick Recap)
Variables
A variable is a letter (x, y, n, p, etc.) that stands for an unknown value. Variables allow us to write general rules instead of listing every specific number.
Algebraic Expressions
An algebraic expression combines numbers, variables, and operations.
- Term: A single number, variable, or product of both (e.g., 3x, 5, -2y).
- Coefficient: The number multiplied by the variable in a term (in 3x, the coefficient is 3).
- Constant: A term with no variable (e.g., 5 in 3x + 5).
- Like terms: Terms with the same variable part (3x and 7x are like terms).
- Unlike terms: Terms with different variable parts (3x and 5y are unlike terms).
Equations
An equation has an equals sign (=) and can be solved.
- LHS = left-hand side, RHS = right-hand side.
- The balancing method: whatever you do to one side, do the same to the other.
- Inverse operations: addition undoes subtraction; multiplication undoes division.
Special Identities
- Additive identity: Adding 0 to any number gives the same number. a + 0 = a.
- Multiplicative identity: Multiplying any number by 1 gives the same number. a x 1 = a.
Exercise 2.1 — Variables and Patterns
Understanding Variables and Matchstick Patterns
Q1. The teacher asks the class to form patterns using matchsticks. The letter L needs 2 matchsticks. How many matchsticks are needed to form n such letters?
- 1 letter L needs 2 matchsticks
- 2 letters L need 4 matchsticks
- 3 letters L need 6 matchsticks
- Pattern: matchsticks = 2 x n
Q2. The letter T needs 2 matchsticks. Write the rule for the number of matchsticks needed to form n such letters.
- 1 letter T = 2 sticks
- 2 letters T = 4 sticks
- 3 letters T = 6 sticks
- Each T adds 2 more sticks
Q3. The letter C needs 3 matchsticks. Write the rule for n such letters.
- 1 C = 3 sticks -> 3 x 1 = 3
- 2 C = 6 sticks -> 3 x 2 = 6
- 3 C = 9 sticks -> 3 x 3 = 9
Q4. The letter Z needs 3 matchsticks. Write the rule.
Q5. The letter V needs 2 matchsticks. The letter E needs 4 matchsticks. The letter N needs 3 matchsticks. Write rules for each.
| Letter | Sticks per letter | Rule |
|---|---|---|
| V | 2 | 2n |
| E | 4 | 4n |
| N | 3 | 3n |
Q6. Observe the pattern: 4, 7, 10, 13, ... Find the general term.
- 1st term: 4 = 3(1) + 1
- 2nd term: 7 = 3(2) + 1
- 3rd term: 10 = 3(3) + 1
- 4th term: 13 = 3(4) + 1
The general term is 3n + 1, where n is the position number.
Q7. Observe the pattern: 1, 4, 9, 16, 25, ... Find the general term.
- 1st term: 1 = 1 squared
- 2nd term: 4 = 2 squared
- 3rd term: 9 = 3 squared
- 4th term: 16 = 4 squared
- 5th term: 25 = 5 squared
The general term is n squared, where n is the position number.
Q8. Observe the pattern: 2, 5, 8, 11, ... Find the general term.
- 1st term: 2 = 3(1) - 1
- 2nd term: 5 = 3(2) - 1
- 3rd term: 8 = 3(3) - 1
- 4th term: 11 = 3(4) - 1
Q9. Meena's age is written as 3 years more than twice Beena's age. If Beena's age is b, express Meena's age.
Q10. A cuboid has length l, breadth b, and height h. Write the formula for its volume.
Exercise 2.2 — Algebraic Expressions and Equations
Terms, Coefficients, and Like/Unlike Terms
Q1. Identify the terms and coefficients in: 5x squared + 3xy - 7
| Term | Coefficient | Type |
|---|---|---|
| 5x sq. | 5 | Variable term |
| 3xy | 3 | Variable term |
| -7 | -- | Constant term |
Q2. Identify like and unlike terms among: 3x, 5y, 7x, -2y, 4x squared
- Like terms (same variable part): 3x and 7x (both have x); 5y and -2y (both have y)
- Unlike terms (different variable part): any term with x vs any term with y vs 4x squared
Q3. Classify the following as monomial, binomial, or trinomial:
(i) 3x (ii) 2x + 5 (iii) 4a - 3b + 7
- (i) 3x -> 1 term -> Monomial
- (ii) 2x + 5 -> 2 terms -> Binomial
- (iii) 4a - 3b + 7 -> 3 terms -> Trinomial
Forming Expressions from Verbal Statements
Q4. Express algebraically:
(i) t added to 100 (ii) 4 less than 9 times y (iii) 3 times x added to 7 (iv) 10 subtracted from m
(i) t added to 100 -> t + 100 (ii) 4 less than 9 times y -> 9y - 4 -> 9y - 4 (iii) 3 times x added to 7 -> 3x + 7 -> 3x + 7 (iv) 10 subtracted from m -> m - 10 -> m - 10
Q5. Express verbally:
(a) 7x + 18 (b) 4x divided by 3
(a) 7x + 18 -> 18 added to 7 times x (or: the sum of 7 times a number and 18) (b) 4x divided by 3 -> 4 times x divided by 3 (or: one-third of 4 times a number)
Q6. Suja's age is a. What will be her age 6 years from now?
Q7. The price of an orange is b rupees. The price of an apple is 50 rupees more than the orange. Express the apple's price.
Q8. In a class, n students like ice cream. What does 2n mean?
Q9. The price of oil per litre is 5 rupees more than 3 times the price of rice per kg. If rice costs p rupees per kg, express the oil price.
Q10. Express algebraically:
(a) y divided by r (b) 10 minus 2 times x
(a) y divided by r -> y/r (or y divided by r) (b) 10 minus 2 times x -> 10 - 2x -> 10 - 2x
Q11. Rajini's father is 5 years more than 3 times Rajini's age. If Rajini's age is x, write the father's age.
Q12. Ramu is 3 years younger than Mathu. If Mathu's age is m, express Ramu's age.
Q13. A square gift box has side p. What length of tape is needed to go around all four sides?
Q14. Find the number of matchsticks needed to form the letter R using the pattern. If one R needs 6 sticks, write the rule for n such letters.
- 1 R = 6 sticks
- 2 R = 12 sticks
- n R = 6n sticks
Q15. Find the rule for the letter S made of 5 matchsticks each.
- 1 S = 5 sticks
- n S = 5n sticks
Q16. A pattern of triangles is made. Each triangle needs 3 matchsticks, but when arranged in a row sharing a side, the pattern is: 1 triangle = 3 sticks, 2 triangles = 5 sticks, 3 triangles = 7 sticks. Find the general rule.
| Triangles | Sticks | Pattern |
|---|---|---|
| 1 | 3 | 2(1) + 1 |
| 2 | 5 | 2(2) + 1 |
| 3 | 7 | 2(3) + 1 |
Exercise 2.3 — Solving Simple Equations
Solving by Balancing Method
Q1. Solve: m + 10 = 16
Q2. Solve: 3x + 10 = 25
Q3. Solve: z / 3 = 4
Step 1: z / 3 = 4
Step 2: Multiply both sides by 3
z = 4 x 3
z = 12
Verification: 12 / 3 = 4 (LHS = RHS)
Q4. Solve: 7y - 20 = 99
Q5. Solve: 2x + 9 = 301
Q6. Solve: p - 5 = 12
Step 1: p - 5 = 12
Step 2: Add 5 to both sides
p = 12 + 5
p = 17
Verification: 17 - 5 = 12 (LHS = RHS)
Q7. Solve: y / 5 = 10
Step 1: y / 5 = 10
Step 2: Multiply both sides by 5
y = 10 x 5
y = 50
Verification: 50 / 5 = 10 (LHS = RHS)
Q8. Solve: 4x - 3 = 17
Step 1: 4x - 3 = 17
Step 2: Add 3 to both sides
4x = 17 + 3
4x = 20
Step 3: Divide both sides by 4
x = 20 / 4
x = 5
Verification: 4(5) - 3 = 20 - 3 = 17 (LHS = RHS)
Q9. Solve: n + 7 = 20
Step 1: n + 7 = 20
Step 2: Subtract 7 from both sides
n = 20 - 7
n = 13
Verification: 13 + 7 = 20 (LHS = RHS)
Q10. Solve: 5a - 8 = 22
Step 1: 5a - 8 = 22
Step 2: Add 8 to both sides
5a = 22 + 8
5a = 30
Step 3: Divide both sides by 5
a = 30 / 5
a = 6
Verification: 5(6) - 8 = 30 - 8 = 22 (LHS = RHS)
Additional Questions
AQ1. What is the additive identity?
a + 0 = a for all a.
AQ2. What is the multiplicative identity?
a x 1 = a for all a.
AQ3. Express algebraically:
(i) t added to 100 (ii) 4 less than 9 times y
(i) t + 100 (ii) 9y - 4
AQ4. Matchstick patterns: Letter R needs 6 sticks, Letter S needs 5 sticks. Write the rules.
- n letters R need 6n matchsticks
- n letters S need 5n matchsticks
AQ5. Suja's age is a. What will be her age 6 years from now?
AQ6. The price of an apple is 50 rupees more than the price of an orange. If the orange costs b rupees, what is the apple's price?
AQ7. In a class, n students like ice cream. What does 2n mean?
AQ8. The price of oil per litre is 5 rupees more than 3 times the price of rice per kg. If rice costs p rupees, express the oil price.
AQ9. Solve: m + 10 = 16
m + 10 = 16
m = 16 - 10
m = 6
Verification: 6 + 10 = 16 (LHS = RHS)
AQ10. Express algebraically:
(a) y divided by r (b) 10 minus 2 times x
(a) y divided by r -> y/r (b) 10 minus 2 times x -> 10 - 2x
AQ11. Express verbally:
(a) 7x + 18 (b) 4x divided by 3
(a) 7x + 18 -> 18 added to 7 times x (b) 4x divided by 3 -> 4 times x divided by 3
AQ12. Rajini's father is 5 years more than 3 times Rajini's age. If Rajini's age is x, write the father's age.
AQ13. A triangular matchstick pattern follows the rule: 1 triangle = 3 sticks, 2 triangles = 5 sticks, 3 triangles = 7 sticks. Find the general rule.
| Triangles (p) | Sticks | Pattern |
|---|---|---|
| 1 | 3 | 2(1)+1 |
| 2 | 5 | 2(2)+1 |
| 3 | 7 | 2(3)+1 |
AQ14. Solve: 3x + 10 = 25
3x + 10 = 25
3x = 25 - 10
3x = 15
x = 15 / 3
x = 5
Verification: 3(5) + 10 = 25 (LHS = RHS)
AQ15. Solve: z / 3 = 4
z / 3 = 4
z = 4 x 3
z = 12
Verification: 12 / 3 = 4 (LHS = RHS)
AQ16. Ramu is 3 years younger than Mathu. If Mathu's age is m, express Ramu's age.
AQ17. Tape needed for a square gift box with side p, going once around all four sides.
AQ18. Solve: 7y - 20 = 99
7y - 20 = 99
7y = 99 + 20
7y = 119
y = 119 / 7
y = 17
Verification: 7(17) - 20 = 119 - 20 = 99 (LHS = RHS)
AQ19. Solve: 2x + 9 = 301
2x + 9 = 301
2x = 301 - 9
2x = 292
x = 292 / 2
x = 146
Verification: 2(146) + 9 = 292 + 9 = 301 (LHS = RHS)
AQ20. Aarthi is 3 years younger than Harini. The sum of their ages is 23. Find Harini's age.
Let Harini's age = h.
Then Aarthi's age = h - 3.
Sum of ages = h + (h - 3) = 23
2h - 3 = 23
2h = 23 + 3
2h = 26
h = 26 / 2
h = 13
Verification: Harini = 13, Aarthi = 13 - 3 = 10. Sum = 13 + 10 = 23 (LHS = RHS)
Common Mistakes
1. Confusing "less than" with subtraction order: "4 less than 9y" means 9y - 4, not 4 - 9y. The quantity after "less than" comes first.
2. Forgetting to do the same operation on both sides: When solving 3x + 5 = 20, some students only subtract 5 from the left side. Always balance -- subtract 5 from both sides.
3. Mixing up expressions and equations: An expression like 3x + 5 cannot be "solved" -- it has no equals sign. Only equations (with =) can be solved for a value.
4. Sign errors when transposing: When moving a term to the other side, its sign changes. +5 becomes -5, and -3 becomes +3. Forgetting this is the most common algebra mistake.
5. Writing 2n as n squared: 2n means "2 times n" (e.g., if n = 3, then 2n = 6). n squared means "n times n" (if n = 3, then n squared = 9). These are not the same.
6. Not verifying the answer: Always substitute your answer back into the original equation. If LHS does not equal RHS, recheck your working.
7. Dividing only one term: When solving 2x + 4 = 14, do NOT divide just the x-term. Subtract 4 first (2x = 10), then divide both sides by 2 (x = 5).
Exam Checklist
Before your exam, make sure you can:
- [ ] Define variable, expression, term, coefficient, and equation correctly
- [ ] Write a matchstick pattern rule from a visual pattern (e.g., letter patterns)
- [ ] Find the general term (nth term) of a number pattern like 3, 7, 11, 15, ...
- [ ] Convert verbal statements to algebraic expressions and vice versa
- [ ] Identify like terms and unlike terms in an expression
- [ ] Classify expressions as monomial, binomial, or trinomial
- [ ] Solve simple linear equations using the balancing method
- [ ] Solve simple linear equations using the trial-and-error method
- [ ] Form equations from word problems and solve them
- [ ] Remember: Additive identity = 0, Multiplicative identity = 1
- [ ] Always verify your answer by substituting back into the original equation