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

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.

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).

**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 $3a^2 + 5ab - 7b + 2$.**

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).

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

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

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

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

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

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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 $E$.

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 $(a + b)^2$?**

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).

**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 $E$.

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

Solution:

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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:

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

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

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.

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: $4x - 2x = 2x$.

• 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.

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