1. The problem is to simplify algebraic expressions by adding like terms. 2. Like terms are terms that have the same variables raised to the same powers. Only like terms can be added or subtracted. 3. Example 1(i): Simplify $4a + 6b + 6 - 2a + b - 3$. Combine like terms: $$4a - 2a = 2a$$ $$6b + b = 7b$$ $$6 - 3 = 3$$ So, the simplified expression is $$2a + 7b + 3$$. 4. Example 1(ii): Simplify $2x^2 - 3x - 7 - x^2 - 5x + 3$. Combine like terms: $$2x^2 - x^2 = x^2$$ $$-3x - 5x = -8x$$ $$-7 + 3 = -4$$ So, the simplified expression is $$x^2 - 8x - 4$$. 5. Exercise 1.1: Simplify each expression by adding like terms. 1. $3x + 4x - 2x = (3 + 4 - 2)x = 5x$ 2. $7a + 3 + 4a + 6 = (7 + 4)a + (3 + 6) = 11a + 9$ 3. $5x + y - 2x + 4y = (5 - 2)x + (1 + 4)y = 3x + 5y$ 4. $5a + 2b - 2a - 4b = (5 - 2)a + (2 - 4)b = 3a - 2b$ 5. $12a + b + 3a + 5b = (12 + 3)a + (1 + 5)b = 15a + 6b$ 6. $3x + 2y + 3 + 4x + 3y + 1 = (3 + 4)x + (2 + 3)y + (3 + 1) = 7x + 5y + 4$ 7. $5x - 4 + 2x + 8 = (5 + 2)x + (-4 + 8) = 7x + 4$ 8. $7x - 4 - 3x + 7 = (7 - 3)x + (-4 + 7) = 4x + 3$ 9. $6a + b + 3 + 2a + 2b - 1 = (6 + 2)a + (1 + 2)b + (3 - 1) = 8a + 3b + 2$ 10. $3x + 4 + 2x - 6 + x + 3 = (3 + 2 + 1)x + (4 - 6 + 3) = 6x + 1$ 11. $3a - b + 4a + 5b - 2a = (3 + 4 - 2)a + (-1 + 5)b = 5a + 4b$ 12. $2ab + 4 + 3ab - 2 = (2 + 3)ab + (4 - 2) = 5ab + 2$ 13. $2p + 3q - r + p - 4q + 2r = (2 + 1)p + (3 - 4)q + (-1 + 2)r = 3p - q + r$ 14. $5k + 3 - 4k + 6 + k - 4 = (5 - 4 + 1)k + (3 + 6 - 4) = 2k + 5$ 15. $2ab + c + 5ab - 4c = (2 + 5)ab + (1 - 4)c = 7ab - 3c$ 16. $3xy + 2z + xy + 9z = (3 + 1)xy + (2 + 9)z = 4xy + 11z$ 17. $6ab + 2cd - ab + 3cd = (6 - 1)ab + (2 + 3)cd = 5ab + 5cd$ 18. $6x - xy + 5x - 7xy = (6 + 5)x + (-1 - 7)xy = 11x - 8xy$ 19. $x^2 - 3x + 4 - 2x^2 + 5x - 3 = (1 - 2)x^2 + (-3 + 5)x + (4 - 3) = -x^2 + 2x + 1$ 20. $3x^2 - 3x + x^2 - 8x + 7 = (3 + 1)x^2 + (-3 - 8)x + 7 = 4x^2 - 11x + 7$ 21. $3a^2 - 2a - 6a + 4a^2 - 3 = (3 + 4)a^2 + (-2 - 6)a - 3 = 7a^2 - 8a - 3$ 22. $y^2 - 8y - 3y^2 + 2y - 3 = (1 - 3)y^2 + (-8 + 2)y - 3 = -2y^2 - 6y - 3$ 23. $3x^2 - 2 + 5x - 4 - 7x + 1 = 3x^2 + (5 - 7)x + (-2 - 4 + 1) = 3x^2 - 2x - 5$ 24. $5a^2 + 2a - 3a^2 + 4 - 3a + 2 = (5 - 3)a^2 + (2 - 3)a + (4 + 2) = 2a^2 - a + 6$ 6. For problem 25, the expression is $3x^2 - 2x + 4xy + 8$. (i) Number of terms: 4 (these are $3x^2$, $-2x$, $4xy$, and $8$). (ii) Coefficient of $xy$ is 4. (iii) Number of variables: 2 (variables are $x$ and $y$). (iv) Constant term is 8. Final answers are the simplified expressions for each exercise and the answers to problem 25.