Algebra worksheet for KS3 and KS4 students focusing on simplifying expressions, expanding brackets, and solving linear equations without a calculator.

Problem: Simplify the following algebraic expressions.

#### Step-by-Step Solutions:

We will simplify each expression by combining like terms. Like terms are terms that have the same variables raised to the same powers.

---

1. \( 5a + 3a \)

Combine the coefficients of \( a \):
\[
5a + 3a = (5 + 3)a = 8a
\]
Answer: \( 8a \)

---

2. \( 6a - 4a \)

Combine the coefficients of \( a \):
\[
6a - 4a = (6 - 4)a = 2a
\]
Answer: \( 2a \)

---

3. \( 4a + a \)

Combine the coefficients of \( a \):
\[
4a + a = (4 + 1)a = 5a
\]
Answer: \( 5a \)

---

4. \( x + x + x + x \)

Combine the coefficients of \( x \):
\[
x + x + x + x = (1 + 1 + 1 + 1)x = 4x
\]
Answer: \( 4x \)

---

5. \( a - a \)

Combine the coefficients of \( a \):
\[
a - a = (1 - 1)a = 0
\]
Answer: \( 0 \)

---

6. \( 3a + 2a - 5a \)

Combine the coefficients of \( a \):
\[
3a + 2a - 5a = (3 + 2 - 5)a = 0
\]
Answer: \( 0 \)

---

7. \( 3a + 5c - a + 2c \)

Combine the coefficients of \( a \) and \( c \) separately:
\[
3a - a = (3 - 1)a = 2a
\]
\[
5c + 2c = (5 + 2)c = 7c
\]
So, the simplified expression is:
\[
2a + 7c
\]
Answer: \( 2a + 7c \)

---

8. \( 3x + 2x + 3y - y \)

Combine the coefficients of \( x \) and \( y \) separately:
\[
3x + 2x = (3 + 2)x = 5x
\]
\[
3y - y = (3 - 1)y = 2y
\]
So, the simplified expression is:
\[
5x + 2y
\]
Answer: \( 5x + 2y \)

---

9. \( 3x - x + 3 - 2 \)

Combine the coefficients of \( x \) and the constants separately:
\[
3x - x = (3 - 1)x = 2x
\]
\[
3 - 2 = 1
\]
So, the simplified expression is:
\[
2x + 1
\]
Answer: \( 2x + 1 \)

---

10. \( 3x + y - x + 4y \)

Combine the coefficients of \( x \) and \( y \) separately:
\[
3x - x = (3 - 1)x = 2x
\]
\[
y + 4y = (1 + 4)y = 5y
\]
So, the simplified expression is:
\[
2x + 5y
\]
Answer: \( 2x + 5y \)

---

11. \( 3x + 2y - 3x + 4y \)

Combine the coefficients of \( x \) and \( y \) separately:
\[
3x - 3x = (3 - 3)x = 0
\]
\[
2y + 4y = (2 + 4)y = 6y
\]
So, the simplified expression is:
\[
6y
\]
Answer: \( 6y \)

---

12. \( 2x + 5y - 3y + x \)

Combine the coefficients of \( x \) and \( y \) separately:
\[
2x + x = (2 + 1)x = 3x
\]
\[
5y - 3y = (5 - 3)y = 2y
\]
So, the simplified expression is:
\[
3x + 2y
\]
Answer: \( 3x + 2y \)

---

13. \( p + q - p - q \)

Combine the coefficients of \( p \) and \( q \) separately:
\[
p - p = (1 - 1)p = 0
\]
\[
q - q = (1 - 1)q = 0
\]
So, the simplified expression is:
\[
0
\]
Answer: \( 0 \)

---

14. \( p + q + p + q + p \)

Combine the coefficients of \( p \) and \( q \) separately:
\[
p + p + p = (1 + 1 + 1)p = 3p
\]
\[
q + q = (1 + 1)q = 2q
\]
So, the simplified expression is:
\[
3p + 2q
\]
Answer: \( 3p + 2q \)

---

15. \( 4p - 5p \)

Combine the coefficients of \( p \):
\[
4p - 5p = (4 - 5)p = -p
\]
Answer: \( -p \)

---

16. \( 5c + 2d - 3c - 4d \)

Combine the coefficients of \( c \) and \( d \) separately:
\[
5c - 3c = (5 - 3)c = 2c
\]
\[
2d - 4d = (2 - 4)d = -2d
\]
So, the simplified expression is:
\[
2c - 2d
\]
Answer: \( 2c - 2d \)

---

17. \( 5x - 3y + 2x - 4y \)

Combine the coefficients of \( x \) and \( y \) separately:
\[
5x + 2x = (5 + 2)x = 7x
\]
\[
-3y - 4y = (-3 - 4)y = -7y
\]
So, the simplified expression is:
\[
7x - 7y
\]
Answer: \( 7x - 7y \)

---

18. \( 5p - 3q + 2 - 4p + 5 + 4q \)

Combine the coefficients of \( p \), \( q \), and the constants separately:
\[
5p - 4p = (5 - 4)p = p
\]
\[
-3q + 4q = (-3 + 4)q = q
\]
\[
2 + 5 = 7
\]
So, the simplified expression is:
\[
p + q + 7
\]
Answer: \( p + q + 7 \)

---

19. \( 2ac + 3ac - 4ac \)

Combine the coefficients of \( ac \):
\[
2ac + 3ac - 4ac = (2 + 3 - 4)ac = 1ac = ac
\]
Answer: \( ac \)

---

20. \( xy + yx \)

Since \( xy \) and \( yx \) are the same term:
\[
xy + yx = 2xy
\]
Answer: \( 2xy \)

---

21. \( 2xy - 4ac + 5yx + 4ac \)

Combine the coefficients of \( xy \) (or \( yx \)) and \( ac \) separately:
\[
2xy + 5yx = 2xy + 5xy = 7xy
\]
\[
-4ac + 4ac = 0
\]
So, the simplified expression is:
\[
7xy
\]
Answer: \( 7xy \)

---

22. \( 3xy + 4xy - xy \)

Combine the coefficients of \( xy \):
\[
3xy + 4xy - xy = (3 + 4 - 1)xy = 6xy
\]
Answer: \( 6xy \)

---

23. \( 3cd - 4cd + cd \)

Combine the coefficients of \( cd \):
\[
3cd - 4cd + cd = (3 - 4 + 1)cd = 0
\]
Answer: \( 0 \)

---

24. \( xy + yx - 2xy + 1 \)

Since \( xy \) and \( yx \) are the same term:
\[
xy + yx - 2xy = xy + xy - 2xy = 2xy - 2xy = 0
\]
So, the simplified expression is:
\[
1
\]
Answer: \( 1 \)

---

25. \( 2ab + 3cd - 4ab - 3cd \)

Combine the coefficients of \( ab \) and \( cd \) separately:
\[
2ab - 4ab = (2 - 4)ab = -2ab
\]
\[
3cd - 3cd = (3 - 3)cd = 0
\]
So, the simplified expression is:
\[
-2ab
\]
Answer: \( -2ab \)

---

26. \( 4y^2 - 3y^2 \)

Combine the coefficients of \( y^2 \):
\[
4y^2 - 3y^2 = (4 - 3)y^2 = y^2
\]
Answer: \( y^2 \)

---

27. \( 4x^3 - x^3 \)

Combine the coefficients of \( x^3 \):
\[
4x^3 - x^3 = (4 - 1)x^3 = 3x^3
\]
Answer: \( 3x^3 \)

---

28. \( 3x^2 + y^3 - x^2 - y^3 \)

Combine the coefficients of \( x^2 \) and \( y^3 \) separately:
\[
3x^2 - x^2 = (3 - 1)x^2 = 2x^2
\]
\[
y^3 - y^3 = (1 - 1)y^3 = 0
\]
So, the simplified expression is:
\[
2x^2
\]
Answer: \( 2x^2 \)

---

29. \( 4y^2 + 5y - 3y^2 - 4y \)

Combine the coefficients of \( y^2 \) and \( y \) separately:
\[
4y^2 - 3y^2 = (4 - 3)y^2 = y^2
\]
\[
5y - 4y = (5 - 4)y = y
\]
So, the simplified expression is:
\[
y^2 + y
\]
Answer: \( y^2 + y \)

---

30. \( 2x^2 + 3x - 5x^2 - x + 8 \)

Combine the coefficients of \( x^2 \), \( x \), and the constant:
\[
2x^2 - 5x^2 = (2 - 5)x^2 = -3x^2
\]
\[
3x - x = (3 - 1)x = 2x
\]
\[
8 = 8
\]
So, the simplified expression is:
\[
-3x^2 + 2x + 8
\]
Answer: \( -3x^2 + 2x + 8 \)

---

31. \( x^2 + x^2 + x^2 \)

Combine the coefficients of \( x^2 \):
\[
x^2 + x^2 + x^2 = (1 + 1 + 1)x^2 = 3x^2
\]
Answer: \( 3x^2 \)

---

32. \( x^2 + x^2 + 3x^2 \)

Combine the coefficients of \( x^2 \):
\[
x^2 + x^2 + 3x^2 = (1 + 1 + 3)x^2 = 5x^2
\]
Answer: \( 5x^2 \)

---

33. \( x^2 + x^2 - x^2 \)

Combine the coefficients of \( x^2 \):
\[
x^2 + x^2 - x^2 = (1 + 1 - 1)x^2 = x^2
\]
Answer: \( x^2 \)

---

34. \( x^2y + xy + x^2y \)

Combine the coefficients of \( x^2y \) and \( xy \) separately:
\[
x^2y + x^2y = (1 + 1)x^2y = 2x^2y
\]
\[
xy = xy
\]
So, the simplified expression is:
\[
2x^2y + xy
\]
Answer: \( 2x^2y + xy \)

---

35. \( x^2y + xy^2 - x^2y + 2y^2x \)

Combine the coefficients of \( x^2y \) and \( xy^2 \) separately:
\[
x^2y - x^2y = (1 - 1)x^2y = 0
\]
\[
xy^2 + 2y^2x = xy^2 + 2xy^2 = (1 + 2)xy^2 = 3xy^2
\]
So, the simplified expression is:
\[
3xy^2
\]
Answer: \( 3xy^2 \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 8a \\
2. & 2a \\
3. & 5a \\
4. & 4x \\
5. & 0 \\
6. & 0 \\
7. & 2a + 7c \\
8. & 5x + 2y \\
9. & 2x + 1 \\
10. & 2x + 5y \\
11. & 6y \\
12. & 3x + 2y \\
13. & 0 \\
14. & 3p + 2q \\
15. & -p \\
16. & 2c - 2d \\
17. & 7x - 7y \\
18. & p + q + 7 \\
19. & ac \\
20. & 2xy \\
21. & 7xy \\
22. & 6xy \\
23. & 0 \\
24. & 1 \\
25. & -2ab \\
26. & y^2 \\
27. & 3x^3 \\
28. & 2x^2 \\
29. & y^2 + y \\
30. & -3x^2 + 2x + 8 \\
31. & 3x^2 \\
32. & 5x^2 \\
33. & x^2 \\
34. & 2x^2y + xy \\
35. & 3xy^2 \\
\end{array}
}
\]