NCERT Solutions for Class 7 Maths Chapter 10: Algebraic Expressions — Free PDF

Chapter 10 Overview: Algebraic Expressions

This chapter introduces you to the formal language of algebra — expressions made up of variables, constants, and arithmetic operations. If you have been solving simple equations in Chapter 4, you already know that variables like $x$ and $y$ represent unknown numbers. Chapter 10 teaches you how to build, classify, and manipulate algebraic expressions systematically.

Algebraic expressions are the building blocks of all higher mathematics. Every equation you will solve in Classes 8, 9, and 10 is built from expressions. Learning to add, subtract, and evaluate them correctly is a skill you will use thousands of times in your math career.

The chapter has 4 exercises with a total of around 20 problems. Exercise 10.1 covers terms, factors, and coefficients. Exercise 10.2 teaches you to identify and group like and unlike terms. Exercise 10.3 focuses on addition and subtraction of expressions. Exercise 10.4 deals with finding the value of an expression by substituting given values for the variables.

Key Concepts and Definitions

Variable: A letter (like $x$, $y$, $a$, $b$) that represents an unknown or changeable number. In the expression $3x + 5$, $x$ is a variable.

Constant: A fixed number that does not change. In $3x + 5$, the number $5$ is a constant. Also, $3$ is a constant multiplied by $x$.

Algebraic Expression: A combination of variables, constants, and operations ($+, -, \times, \div$). Examples: $3x + 5$, $2a^2 - 7b + 1$, $\dfrac{x}{2} + 3y$.

Term: Each part of an expression separated by $+$ or $-$ signs. In $4x^2 - 3xy + 7$, the terms are $4x^2$, $-3xy$, and $7$.

Factor: The numbers and variables that are multiplied together to form a term. In $5xy$, the factors are $5$, $x$, and $y$.

Coefficient: The numerical part of a term. In $-7x^2y$, the coefficient is $-7$. In $x$ (which is $1 \cdot x$), the coefficient is $1$. The coefficient includes the sign.

Like Terms: Terms that have the same variable parts (same variables raised to the same powers). Examples: $3x^2$ and $-5x^2$; $7ab$ and $2ab$. Only like terms can be added or subtracted.

Unlike Terms: Terms with different variable parts. Examples: $3x$ and $3x^2$; $5xy$ and $5xz$.

Types of Expressions:
- Monomial: $1$ term (e.g., $5x^2$, $-3ab$, $7$)
- Binomial: $2$ terms (e.g., $3x + 7$, $a^2 - b^2$)
- Trinomial: $3$ terms (e.g., $x^2 + 3x + 2$)
- Polynomial: A general expression with one or more terms (monomials, binomials, and trinomials are all types of polynomials).

Exercise 10.1 — Terms, Factors, and Coefficients (Solved)

Q1. Identify the terms in $4x^2 - 3xy + 5y + 7$.

The terms are: $4x^2$, $-3xy$, $5y$, and $7$.

Note: The sign belongs to the term. The second term is $-3xy$ (not just $3xy$).

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Q2. Identify the coefficient of $x$ in each:

(a) $5x + 3$ — coefficient of $x$ is $5$.

(b) $-7x^2 + 2x$ — coefficient of $x$ is $2$; coefficient of $x^2$ is $-7$.

(c) $x + 1$ — coefficient of $x$ is $1$ (often not written explicitly, but it is there).

(d) $-x + 4$ — coefficient of $x$ is $-1$.

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Q3. Classify the following as monomials, binomials, or trinomials:

- $5x^2$ — Monomial ($1$ term)
- $3x + 7$ — Binomial ($2$ terms)
- $4x^2 - 3x + 1$ — Trinomial ($3$ terms)
- $2xy$ — Monomial
- $a + b + c + d$ — Polynomial with $4$ terms (none of the specific names apply)

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Q4. Write two like terms for $3x^2y$.

Like terms have the same variable part. Examples: $-5x^2y$ and $x^2y$ (or $\dfrac{1}{2}x^2y$).

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Q5. Identify the terms and their factors in $3a^2 + 5ab - 7b + 2$.

| Term | Factors |
|---|---|
| $3a^2$ | $3, a, a$ |
| $5ab$ | $5, a, b$ |
| $-7b$ | $-7, b$ |
| $2$ | $2$ (constant) |

Exercise 10.2 — Like and Unlike Terms (Solved)

Like terms have the same variables raised to the same powers. Only the coefficients can differ.

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Q1. Group the like terms:
$3x, \; 5y, \; -2x, \; 7, \; -3y, \; 4x, \; 10$

Group 1 ($x$ terms): $3x, -2x, 4x$

Group 2 ($y$ terms): $5y, -3y$

Group 3 (constants): $7, 10$

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Q2. Simplify by combining like terms: $7x - 3x + 5x - x$.

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Q3. Simplify: $3x^2 + 5x - 2x^2 + x - 4$.

Combine $x^2$ terms: $3x^2 - 2x^2 = x^2$.

Combine $x$ terms: $5x + x = 6x$.

Constants: $-4$.

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Q4. Are $5xy$ and $5yx$ like terms?

Yes! $xy = yx$ (multiplication is commutative), so $5xy$ and $5yx$ are like terms. They can be combined.

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Q5. Are $3x^2$ and $3x$ like terms?

No! The powers of $x$ are different ($x^2$ vs $x$). These are unlike terms and cannot be combined.

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Q6. Simplify: $2a^2b + 3ab^2 - a^2b + 5ab^2$.

$a^2b$ terms: $2a^2b - a^2b = a^2b$.

$ab^2$ terms: $3ab^2 + 5ab^2 = 8ab^2$.

Note: $a^2b$ and $ab^2$ are NOT like terms (different variable parts).

Exercise 10.3 — Addition and Subtraction of Expressions (Solved)

Q1. Add $3x + 5$ and $4x - 2$.

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Q2. Add $2x^2 + 3x + 1$ and $x^2 - 5x + 4$.

$$= (2 + 1)x^2 + (3 - 5)x + (1 + 4)$$

$$= 3x^2 - 2x + 5$$

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Q3. Subtract $2x^2 - 3x + 1$ from $5x^2 + 4x - 7$.

$$= 5x^2 + 4x - 7 - 2x^2 + 3x - 1$$

$$= 3x^2 + 7x - 8$$

Important: When subtracting, change the sign of every term in the expression being subtracted.

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Q4. Subtract $3a + 4b - 2c$ from $7a - b + 5c$.

$$= 7a - b + 5c - 3a - 4b + 2c$$

$$= 4a - 5b + 7c$$

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Q5. Add: $5p^2 + 3p - 7$, $2p^2 - p + 4$, and $-p^2 + 6p - 1$.

Using the column method:

$p^2$ terms: $5p^2 + 2p^2 + (-p^2) = 6p^2$

$p$ terms: $3p + (-p) + 6p = 8p$

Constants: $-7 + 4 + (-1) = -4$

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Q6. What should be added to $3x^2 - 5x + 2$ to get $7x^2 + x - 6$?

Let the required expression be $E$.

$$E = (7x^2 + x - 6) - (3x^2 - 5x + 2)$$

$$= 7x^2 + x - 6 - 3x^2 + 5x - 2 = 4x^2 + 6x - 8$$

Exercise 10.4 — Finding the Value of an Expression (Solved)

To find the value of an expression, substitute the given values of the variables and evaluate using BODMAS.

Q1. Find the value of $3x + 2$ when $x = 4$.

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Q2. Find the value of $x^2 + 2x - 3$ when $x = -2$.

Note: $(-2)^2 = 4$ (positive), not $-4$. Be very careful with negative numbers!

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Q3. Find the value of $2a^2b + 3ab$ when $a = 1$ and $b = -1$.

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Q4. If $p = -2$, find the value of $4p^2 - 3p + 7$.

Note: $-3 \times (-2) = +6$ (negative times negative is positive).

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Q5. Find the value of $\dfrac{x^2 - 1}{x + 1}$ when $x = 3$.

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Q6. If $a = 2$ and $b = 3$, find the value of $a^2 + b^2 + 2ab$. Does it equal $(a + b)^2$?

$$(a + b)^2 = (2 + 3)^2 = 5^2 = 25$$

Yes, they are equal! This is because $a^2 + 2ab + b^2 = (a + b)^2$ is an algebraic identity (you will study this formally in Class 8, Chapter 8).

Additional Worked Examples

Example 1. The perimeter of a rectangle with length $l$ and breadth $b$ is $2(l + b)$. If $l = 3x + 2$ and $b = x - 1$, find the perimeter.

Solution:

If $x = 5$: $P = 8(5) + 2 = 42$ units.

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Example 2. What should be subtracted from $5a^2 - 3a + 9$ to get $2a^2 + a - 1$?

Solution:
Let the expression to be subtracted be $E$.

$$E = (5a^2 - 3a + 9) - (2a^2 + a - 1)$$

$$= 5a^2 - 3a + 9 - 2a^2 - a + 1 = 3a^2 - 4a + 10$$

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Example 3. If $x = 2$ and $y = -3$, evaluate $3x^2y - 2xy^2 + xy$.

Solution:

$$= -36 - 36 + (-6)$$

$$= -78$$

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Example 4. Simplify and then find the value for $x = 1$: $(4x^2 + 3x - 2) + (x^2 - 5x + 6) - (2x^2 + x - 3)$.

Solution:
Simplify first:

$x^2$ terms: $4 + 1 - 2 = 3x^2$.
$x$ terms: $3 - 5 - 1 = -3x$.
Constants: $-2 + 6 + 3 = 7$.

Result $= 3x^2 - 3x + 7$.

For $x = 1$: $3(1) - 3(1) + 7 = 3 - 3 + 7 = 7$.

Common Mistakes to Avoid

Mistake 1: Forgetting to change ALL signs when subtracting.
When subtracting $(2x^2 - 3x + 1)$, you must change ALL three signs: $-2x^2 + 3x - 1$. Missing even one sign leads to a wrong answer.

Mistake 2: Combining unlike terms.
$3x^2 + 2x \neq 5x^3$ or $5x^2$. You cannot combine $x^2$ and $x$ terms — they are unlike terms. Similarly, $5xy + 3x \neq 8xy$.

Mistake 3: Sign errors with negative numbers.
$(-2)^2 = 4$ (positive), NOT $-4$. Also, $-3 \times (-2) = +6$, not $-6$. Be especially careful when substituting negative values.

Mistake 4: Forgetting the coefficient $1$ or $-1$.
In the term $x$, the coefficient is $1$ (not zero). In $-ab$, the coefficient is $-1$. These hidden $1$s are easy to forget.

Mistake 5: Treating $x^2$ and $2x$ as like terms.
The variable part must be EXACTLY the same. $x^2$ and $x$ have different powers, so they are unlike terms. $3xy$ and $3x^2y$ are also unlike terms.

Exam Tips for Algebraic Expressions

Practice Questions with Answers

Q1. Add: $5a - 3b + 2$ and $-2a + 7b - 4$.

Answer: $(5a - 2a) + (-3b + 7b) + (2 - 4) = 3a + 4b - 2$.

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Q2. Subtract $x^2 + 3x - 5$ from $4x^2 - x + 2$.

Answer: $(4x^2 - x + 2) - (x^2 + 3x - 5) = 4x^2 - x + 2 - x^2 - 3x + 5 = 3x^2 - 4x + 7$.

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Q3. Find the value of $2x^2 - 3x + 1$ when $x = -1$.

Answer: $2(-1)^2 - 3(-1) + 1 = 2 + 3 + 1 = 6$.

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Q4. What should be added to $2p - 3q + 7$ to get $5p + q - 2$?

Answer: $(5p + q - 2) - (2p - 3q + 7) = 3p + 4q - 9$.

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Q5. Simplify: $4x - [3x - (2x - x)]$.

Answer: Inner bracket: $2x - x = x$. Then: $3x - x = 2x$. Finally: $4x - 2x = 2x$.

Key Takeaways

- Term: A product of factors (e.g., $3x^2y$ is a single term).
- Coefficient: The numerical part of a term, including the sign ($-7$ in $-7x^2$).
- Like terms: Same variable parts ($3x$ and $-5x$). Unlike terms: different variable parts ($3x$ and $3x^2$).
- Monomial $= 1$ term, Binomial $= 2$ terms, Trinomial $= 3$ terms.
- Addition: Combine like terms by adding their coefficients.
- Subtraction: Change the sign of every term being subtracted, then add.
- Finding value: Substitute the given values and evaluate using BODMAS. Be careful with negative numbers.
- This chapter lays the groundwork for Chapter 8 (Algebraic Expressions and Identities) in Class 8.

Practice algebraic expressions on SparkEd's interactive module for instant feedback!

Ten Common Mistakes Students Make in Algebraic Expressions

This chapter is where algebra officially begins, and where students first start losing marks to silly sign mistakes. Here are the ten errors that come up most in CBSE Class 7 exam sheets.

Mistake 1: Combining unlike terms
❌ Wrong: $3x^2 + 2x = 5x^3$ or $5x^2$.
✅ Correct: $3x^2 + 2x$ cannot be simplified further. $x^2$ and $x$ are different variable parts.

Mistake 2: Forgetting to flip signs when subtracting
❌ Wrong: $(5x + 3) - (2x - 1) = 5x + 3 - 2x - 1 = 3x + 2$.
✅ Correct: $(5x + 3) - (2x - 1) = 5x + 3 - 2x + 1 = 3x + 4$.
Every sign of the expression being subtracted must flip.

Mistake 3: Treating $(-2)^2$ as $-4$
❌ Wrong: $(-2)^2 = -4$.
✅ Correct: $(-2)^2 = 4$.
A negative squared is positive because negative times negative is positive.

Mistake 4: Forgetting the coefficient $1$ in front of a term
❌ Wrong: The coefficient of $x$ in $x + 5$ is $0$.
✅ Correct: The coefficient of $x$ is $1$.
Similarly the coefficient of $-x$ is $-1$, not $0$.

Mistake 5: Confusing like terms with same variables
❌ Wrong: $3xy$ and $3x^2y$ are like terms because both have $x$ and $y$.
✅ Correct: They are unlike because the powers of $x$ differ.
Like terms must have the same variables with the same powers.

Mistake 6: Wrong use of BODMAS during substitution
❌ Wrong: For $x = 2$, writing $2x + 3 \times 4 = 5 \times 4 = 20$.
✅ Correct: $2x + 3 \times 4 = 4 + 12 = 16$.
Multiplication is evaluated before addition.

Mistake 7: Distributing incorrectly
❌ Wrong: $-2(x - 3) = -2x - 6$.
✅ Correct: $-2(x - 3) = -2x + 6$.
The negative sign multiplies both terms inside the bracket.

Mistake 8: Writing the coefficient without its sign
❌ Wrong: The coefficient of $y$ in $4x - 7y$ is $7$.
✅ Correct: The coefficient is $-7$.

Mistake 9: Treating a constant as a variable
❌ Wrong: Saying the expression $5$ is a monomial in $x$.
✅ Correct: $5$ is a constant. It is still a monomial but it does not depend on $x$.

Mistake 10: Not checking substitution with a quick value
❌ Wrong: Writing a simplified expression without verifying.
✅ Correct: Substitute $x = 1$ into both the original and simplified expression. They should give the same number.

Previous Year Questions (PYQs) with Solutions

These are CBSE Class 7 style Algebraic Expressions problems from recent years.

Q1. Identify the like terms in the list: $3x, 5xy, -2x, 7y, -4xy$. `(CBSE 2021, 2 marks)`
Solution: $3x$ and $-2x$ are like. $5xy$ and $-4xy$ are like.
Answer: Two pairs.

Q2. Add: $4a^2 - 3a + 7$ and $2a^2 + 5a - 2$. `(CBSE 2022, 2 marks)`
Solution: $(4a^2 + 2a^2) + (-3a + 5a) + (7 - 2) = 6a^2 + 2a + 5$.
Answer: $6a^2 + 2a + 5$

Q3. Subtract $3x^2 - 4x + 9$ from $6x^2 + 2x - 5$. `(CBSE 2023, 3 marks)`
Solution:
$(6x^2 + 2x - 5) - (3x^2 - 4x + 9) = 6x^2 + 2x - 5 - 3x^2 + 4x - 9 = 3x^2 + 6x - 14$.
Answer: $3x^2 + 6x - 14$

Q4. Find the value of $2x^2 - 5x + 3$ when $x = -1$. `(CBSE 2022, 2 marks)`
Solution: $2(-1)^2 - 5(-1) + 3 = 2 + 5 + 3 = 10$.
Answer: $10$

Q5. What should be added to $x^2 + 3x - 5$ to get $4x^2 - x + 7$? `(CBSE 2023, 3 marks)`
Solution:
$(4x^2 - x + 7) - (x^2 + 3x - 5) = 3x^2 - 4x + 12$.
Answer: $3x^2 - 4x + 12$

Q6. Simplify: $3(a + b) - 2(a - b) + 4b$. `(CBSE 2021, 3 marks)`
Solution: $3a + 3b - 2a + 2b + 4b = a + 9b$.
Answer: $a + 9b$

Q7. Find the value of $x^2 + y^2 - 2xy$ when $x = 3$ and $y = -2$. `(CBSE 2024, 3 marks)`
Solution: $9 + 4 - 2(3)(-2) = 9 + 4 + 12 = 25$.
Answer: $25$

Q8 (HOTS). The perimeter of a triangle is $(7x - 3)$ cm. Two of its sides are $(2x + 1)$ cm and $(x + 5)$ cm. Find the third side. `(CBSE 2024, 4 marks)`
Solution: Third side $= (7x - 3) - (2x + 1) - (x + 5) = 7x - 3 - 2x - 1 - x - 5 = 4x - 9$.
Answer: $(4x - 9)$ cm

Practice This Chapter on SparkEd

Algebraic Expressions is the start of algebra proper, which means every small habit you build here will either help or hurt you for the next four years of school maths. The SparkEd Algebraic Expressions module is designed to build the right habits from day one.

The module has over 60 problems across three levels. Level 1 focuses on basic skills: identifying like terms, adding and subtracting monomials, and finding coefficients. Level 2 moves into simplification of multi term expressions, substitution problems, and removing brackets with the correct sign. Level 3 is the CBSE exam zone with multi step simplifications, word problems that produce expressions, and HOTS style questions involving perimeters and formulas.

The instant AI feedback is especially helpful because this chapter has so many sign errors. The AI looks at your working and shows the exact step where a sign went wrong or a term was incorrectly combined. Gamification keeps you motivated and the adaptive engine targets your weak spots.

Practice Algebraic Expressions on SparkEd →

Download Algebraic Expressions worksheet

Aim for twenty five minutes a day for a week. By the end you will simplify expressions without hesitation.