11+ Maths: Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables (letters that represent unknown values), and arithmetic operations such as addition, subtraction, multiplication, and division. Understanding algebraic expressions is crucial for solving problems in mathematics, and it is a key component of the 11+ exam.

Key Concepts in Algebraic Expressions

1. Components of Algebraic Expressions

• Terms: The parts of an expression separated by addition or subtraction. For example, in the expression $$3x + 4y – 5$$, the terms are $$3x$$, $$4y$$, and $$-5$$.
• Coefficients: The numerical factor in a term. In the term $$3x$$, 3 is the coefficient.
• Variables: The letters that represent unknown values. In $$3x$$, $$x$$ is the variable.
• Constants: Fixed values that do not change. In the expression $$3x + 4y – 5$$, -5 is a constant.

2. Simplifying Expressions

To simplify an algebraic expression, combine like terms (terms that have the same variable raised to the same power). For example:
$$2x + 3x = 5x$$

3. Evaluating Expressions

To evaluate an algebraic expression, substitute the values of the variables and perform the arithmetic operations. For example, to evaluate $$2x + 3$$ when $$x = 4$$:
$$2(4) + 3 = 8 + 3 = 11$$

4. Operations with Algebraic Expressions

• Addition: Combine like terms.
• Subtraction: Distribute the negative sign and then combine like terms.
• Multiplication: Use the distributive property (e.g., $$a(b + c) = ab + ac$$).
• Division: Simplify the expression by dividing the coefficients and maintaining the variables.

5. Algebraic Identities

Certain algebraic identities can help simplify expressions:

• $$ (a + b)^2 = a^2 + 2ab + b^2 $$
• $$ (a – b)^2 = a^2 – 2ab + b^2 $$
• $$ a^2 – b^2 = (a + b)(a – b) $$

Practice Questions on Algebraic Expressions

Easy Level

1. Simplify: $$2x + 3x$$.
2. What is the value of $$3a + 4$$ when $$a = 2$$?
3. Combine the like terms: $$5y + 2y – 3y$$.
4. Simplify: $$4m + 6 – 2m$$.
5. Evaluate: $$x^2 + 3$$ when $$x = 3$$.
6. What is the coefficient of $$x$$ in the expression $$5x + 2$$?
7. Simplify: $$7p – 2p + 3$$.
8. What is the value of $$4x – 5$$ when $$x = 3$$?
9. Combine the like terms: $$10k + 5 – 3k$$.
10. Simplify: $$8a + 4 – 2a$$.
11. What is the constant term in the expression $$6x + 7$$?
12. Evaluate: $$2y + 4$$ when $$y = 1$$.
13. Simplify: $$3x + 5x$$.
14. What is the value of $$x + 2$$ when $$x = 5$$?
15. Combine the like terms: $$2x + 3x + 4$$.
16. Simplify: $$9b – 3b + 6$$.
17. What is the coefficient of $$y$$ in the expression $$4y – 5$$?
18. Evaluate: $$x^2 + 2x + 1$$ when $$x = 2$$.
19. Combine the like terms: $$5x + 2x – x$$.
20. Simplify: $$6a + 2b – 3a$$.

Medium Level

1. Simplify: $$3x + 4y – 2x + y$$.
2. What is the value of $$2a + 3b$$ when $$a = 2$$ and $$b = 3$$?
3. Combine the like terms: $$5m + 2n – 3m + 4n$$.
4. Simplify: $$8p – 3 – 2p + 5$$.
5. Evaluate: $$4x – 3x + 2$$ when $$x = 4$$.
6. If $$x = 5$$, what is the value of $$x^2 + 2x – 1$$?
7. Simplify: $$6a + 4 – 2a + 3$$.
8. Combine the like terms: $$7x + 2y – 4x + 3y$$.
9. What is the coefficient of $$z$$ in the expression $$3z + 4z – 2$$?
10. Evaluate: $$2a^2 + 3a$$ when $$a = 3$$.
11. Simplify: $$9x + 5 – 3x + 7$$.
12. Combine the like terms: $$4m + 2n – m + n$$.
13. What is the value of $$3y – 2$$ when $$y = 6$$?
14. Simplify: $$5x – 2(3 – x)$$.
15. If $$a = 2$$ and $$b = 3$$, what is the value of $$ab + a^2$$?
16. Evaluate: $$3x + 4y – 2x$$ when $$x = 2$$ and $$y = 3$$.
17. Combine the like terms: $$5k + 4 – 2k + 3$$.
18. What is the constant term in the expression $$8m – 3 + 4$$?
19. Simplify: $$4p + 2 – 3p + 5$$.
20. What is the value of $$x^2 + 5x + 6$$ when $$x = 1$$?

Hard Level

1. Simplify: $$2x + 3(4 – x)$$.
2. What is the value of $$5a + 2b – 3a$$ when $$a = 4$$ and $$b = 2$$?
3. Combine the like terms: $$2x + 3y – x + 4y$$.
4. Simplify: $$10m – 2(3m – 4)$$.
5. Evaluate: $$2x^2 + 3x + 5$$ when $$x = 2$$.
6. If $$y = 3$$, what is the value of $$2y^2 + 4y – 5$$?
7. Simplify: $$6(a + 2) – 4a$$.
8. Combine the like terms: $$7x – 3(2 – x)$$.
9. What is the coefficient of $$x^2$$ in the expression $$3x^2 + 4x – 5$$?
10. Evaluate: $$5(x + 2) – 3(x – 4)$$ when $$x = 1$$.
11. Simplify: $$8x – 2(2x + 5)$$.
12. If $$x = 3$$, what is the value of $$3x^2 + 2x – 1$$?
13. Combine the like terms: $$4p + 5 – 2(p + 3)$$.
14. What is the value of $$4(a – 3) + 2$$ when $$a = 6$$?
15. Simplify: $$3(x + 4) – 2(x – 2)$$.
16. If $$z = 2$$, what is the value of $$z^2 + 3z + 1$$?
17. Combine the like terms: $$6a + 4b – 2a + 3b$$.
18. What is the coefficient of $$y$$ in the expression $$7y – 4y + 9$$?
19. Simplify: $$5(2x – 3) + 3(x + 2)$$.
20. Evaluate: $$2x^2 + 4x – 6$$ when $$x = -2$$.

Answers and Explanations

Easy Level

1. $$ 2x + 3x = 5x $$
2. $$ 3(2) + 4 = 6 + 4 = 10 $$
3. $$ 5y + 2y – 3y = 4y $$
4. $$ 4m + 6 – 2m = 2m + 6 $$
5. $$ x^2 + 3 = 3^2 + 3 = 9 + 3 = 12 $$
6. $$ 5 $$ (coefficient of $$ x $$)
7. $$ 7p – 2p + 3 = 5p + 3 $$
8. $$ 4(3) – 5 = 12 – 5 = 7 $$
9. $$ 10k +

5 – 3k = 7k + 5 $$

10. $$ 8a + 4 – 2a = 6a + 4 $$
11. $$ 0.25 $$ (1/4)
12. $$ 1/5 $$
13. $$ 3 $$ (constant term)
14. $$ 2(1) + 4 = 6 $$
15. $$ 2 $$ (2/5)
16. $$ 5x + 2 = 8x $$
17. $$ 1.75 $$ (7/4)
18. $$ 2 $$ (1/2)
19. $$ 3 $$
20. $$ 8 $$ (1/8)

Medium Level

1. $$ 3x + 4y – 2x + y = x + 5y $$
2. $$ 2(2) + 3(3) = 4 + 9 = 13 $$
3. $$ 5m + 2n – 3m + 4n = 2m + 6n $$
4. $$ 8p – 3 – 2p + 5 = 6p + 2 $$
5. $$ 4(4) – 3(2) + 2 = 8 $$
6. $$ 7(3) + 4 = 21 + 4 = 25 $$
7. $$ 6a + 4 – 2a + 3 = 4a + 7 $$
8. $$ 7x + 2y – 4x + 3y = 3x + 5y $$
9. $$ 5 $$ (coefficient of $$ z $$)
10. $$ 3(2) + 4(3) – 1 = 6 + 12 – 1 = 17 $$
11. $$ 9x – 3 – 2(4) = 3 + 2 $$
12. $$ 1.5 $$ (3/2)
13. $$ 2 $$ (5)
14. $$ 2a + 3 + 5 = 2a + 8 $$
15. $$ 3x + 12 – 4x = -x + 12 $$
16. $$ 5 $$ (7)
17. $$ 2 $$ (1)
18. $$ 8 $$ (8)
19. $$ 3 $$
20. $$ 7 $$ (7)

Hard Level

1. $$ 2x + 3(4 – x) = 2x + 12 – 3x = -x + 12 $$
2. $$ 5(4) + 2(3) – 3(2) = 20 + 6 – 6 = 20 $$
3. $$ 4x + 3y – 2x + 4y = 2x + 7y $$
4. $$ 10m – 2(3m – 4) = 10m – 6m + 8 = 4m + 8 $$
5. $$ 2x + 3x + 5 = 5x + 5 $$
6. $$ 2(2) + 4(3) + 1 = 4 + 12 + 1 = 17 $$
7. $$ 6(a + 2) – 4a = 6a + 12 – 4a = 2a + 12 $$
8. $$ 7x – 6 = 6x + 6 $$
9. $$ 5(2x + 4) + 3(x + 2) = 10x + 20 + 3x + 6 = 13x + 26 $$
10. $$ 4a + 3a = 7a $$
11. $$ 5x – 3 = 0 $$
12. $$ 0 $$
13. $$ 5(2) = 0 $$
14. $$ 3(2) + 5 = 11 $$
15. $$ 6 $$
16. $$ 1 $$ (4)
17. $$ 6x + 12 – 2(2x) = 2x + 12 $$
18. $$ 0.1 + 0.2 + 0.3 = 0.6 $$
19. $$ 2 $$
20. $$ 4 $$

This set of questions and answers provides a comprehensive overview of algebraic expressions relevant to the 11+ exam, covering various difficulty levels and encouraging students to develop their understanding and problem-solving skills.