Mixed Basic Algebra - Worksheet - Free Printable

Algebra Homework Solution

The homework is divided into several sections: Simplifying, Expanding Brackets, Factorising, and Substituting. Below, I will solve each problem step by step.

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## Simplifying

1. \( a + 2a + a \)
- Combine like terms:
\[
a + 2a + a = 4a
\]
- Answer: \( 4a \)

2. \( 4b + 2b + 6b \)
- Combine like terms:
\[
4b + 2b + 6b = 12b
\]
- Answer: \( 12b \)

3. \( 7c - 2c + 6d + 8d \)
- Combine like terms for \( c \) and \( d \):
\[
7c - 2c = 5c \quad \text{and} \quad 6d + 8d = 14d
\]
\[
7c - 2c + 6d + 8d = 5c + 14d
\]
- Answer: \( 5c + 14d \)

4. \( 11e + 6f - 9e - 3f \)
- Combine like terms for \( e \) and \( f \):
\[
11e - 9e = 2e \quad \text{and} \quad 6f - 3f = 3f
\]
\[
11e + 6f - 9e - 3f = 2e + 3f
\]
- Answer: \( 2e + 3f \)

5. \( 3g - 4h + 3 - 3g - 4h + 4 \)
- Combine like terms for \( g \), \( h \), and constants:
\[
3g - 3g = 0, \quad -4h - 4h = -8h, \quad 3 + 4 = 7
\]
\[
3g - 4h + 3 - 3g - 4h + 4 = -8h + 7
\]
- Answer: \( -8h + 7 \)

6. \( 2 + 3x + 4x^2 + 3x + 2 \)
- Combine like terms for \( x^2 \), \( x \), and constants:
\[
4x^2 \quad \text{(no other \( x^2 \) terms)}
\]
\[
3x + 3x = 6x
\]
\[
2 + 2 = 4
\]
\[
2 + 3x + 4x^2 + 3x + 2 = 4x^2 + 6x + 4
\]
- Answer: \( 4x^2 + 6x + 4 \)

7. Spot the errors in the following simplified expressions:
- a) \( 3a + 8a - 5a = 16a \)
- Correct simplification:
\[
3a + 8a - 5a = (3 + 8 - 5)a = 6a
\]
- Error: The result should be \( 6a \), not \( 16a \).

- b) \( 4bc + 5b - 3c = 6bc \)
- Correct simplification:
\[
4bc, 5b, \text{and } -3c \text{ are not like terms and cannot be combined.}
\]
- Error: The terms \( 4bc \), \( 5b \), and \( -3c \) are not like terms and cannot be simplified to \( 6bc \).

- c) \( x^2 + 3x + 4x + 2 + 1 = 8x + 3 \)
- Correct simplification:
\[
x^2 \quad \text{(no other \( x^2 \) terms)}
\]
\[
3x + 4x = 7x
\]
\[
2 + 1 = 3
\]
\[
x^2 + 3x + 4x + 2 + 1 = x^2 + 7x + 3
\]
- Error: The result should be \( x^2 + 7x + 3 \), not \( 8x + 3 \).

---

## Expanding Brackets

1. \( 4(a + 3) \)
- Distribute the 4:
\[
4(a + 3) = 4a + 12
\]
- Answer: \( 4a + 12 \)

2. \( 6(b - 5) \)
- Distribute the 6:
\[
6(b - 5) = 6b - 30
\]
- Answer: \( 6b - 30 \)

3. \( 7(2c + 3) \)
- Distribute the 7:
\[
7(2c + 3) = 14c + 21
\]
- Answer: \( 14c + 21 \)

4. \( 5(2d + 3e) \)
- Distribute the 5:
\[
5(2d + 3e) = 10d + 15e
\]
- Answer: \( 10d + 15e \)

5. \( f(3g + 2) \)
- Distribute \( f \):
\[
f(3g + 2) = 3fg + 2f
\]
- Answer: \( 3fg + 2f \)

6. \( 2h(h - 4) \)
- Distribute \( 2h \):
\[
2h(h - 4) = 2h^2 - 8h
\]
- Answer: \( 2h^2 - 8h \)

7. \( 4(2x + 3) + 3(3x + 2) \)
- Distribute and combine like terms:
\[
4(2x + 3) = 8x + 12
\]
\[
3(3x + 2) = 9x + 6
\]
\[
4(2x + 3) + 3(3x + 2) = (8x + 12) + (9x + 6) = 17x + 18
\]
- Answer: \( 17x + 18 \)

8. \( 5(3y - 4) + 2(6 - 5y) \)
- Distribute and combine like terms:
\[
5(3y - 4) = 15y - 20
\]
\[
2(6 - 5y) = 12 - 10y
\]
\[
5(3y - 4) + 2(6 - 5y) = (15y - 20) + (12 - 10y) = 5y - 8
\]
- Answer: \( 5y - 8 \)

---

## Factorising

1. \( 2x + 8 \)
- Factor out the greatest common factor (GCF), which is 2:
\[
2x + 8 = 2(x + 4)
\]
- Answer: \( 2(x + 4) \)

2. \( 3x + 9 \)
- Factor out the GCF, which is 3:
\[
3x + 9 = 3(x + 3)
\]
- Answer: \( 3(x + 3) \)

3. \( 6x - 8 \)
- Factor out the GCF, which is 2:
\[
6x - 8 = 2(3x - 4)
\]
- Answer: \( 2(3x - 4) \)

4. \( 11x + 88 \)
- Factor out the GCF, which is 11:
\[
11x + 88 = 11(x + 8)
\]
- Answer: \( 11(x + 8) \)

5. \( 8x - 12 \)
- Factor out the GCF, which is 4:
\[
8x - 12 = 4(2x - 3)
\]
- Answer: \( 4(2x - 3) \)

6. \( 24x + 40x \)
- Combine like terms first:
\[
24x + 40x = 64x
\]
- Factor out the GCF, which is 16:
\[
64x = 16(4x)
\]
- Answer: \( 16(4x) \)

7. \( 7x + 14xy \)
- Factor out the GCF, which is \( 7x \):
\[
7x + 14xy = 7x(1 + 2y)
\]
- Answer: \( 7x(1 + 2y) \)

8. \( 12xy - 16x \)
- Factor out the GCF, which is \( 4x \):
\[
12xy - 16x = 4x(3y - 4)
\]
- Answer: \( 4x(3y - 4) \)

9. \( 10x^2 + 15x \)
- Factor out the GCF, which is \( 5x \):
\[
10x^2 + 15x = 5x(2x + 3)
\]
- Answer: \( 5x(2x + 3) \)

---

## Substituting

1. Find \( 5x \) when:
- a) \( x = 4 \)
\[
5x = 5(4) = 20
\]
- b) \( x = 30 \)
\[
5x = 5(30) = 150
\]
- c) \( x = -3 \)
\[
5x = 5(-3) = -15
\]

- Answers:
- a) \( 20 \)
- b) \( 150 \)
- c) \( -15 \)

2. Find \( 3x - 2y \) when:
- a) \( x = 3 \), \( y = 1 \)
\[
3x - 2y = 3(3) - 2(1) = 9 - 2 = 7
\]
- b) \( x = 5 \), \( y = -2 \)
\[
3x - 2y = 3(5) - 2(-2) = 15 + 4 = 19
\]
- c) \( x = -2 \), \( y = 3 \)
\[
3x - 2y = 3(-2) - 2(3) = -6 - 6 = -12
\]

- Answers:
- a) \( 7 \)
- b) \( 19 \)
- c) \( -12 \)

3. When \( f = 3 \), \( g = 4 \), \( h = 2 \), find:
- a) \( 3f + 2g \)
\[
3f + 2g = 3(3) + 2(4) = 9 + 8 = 17
\]
- b) \( 5gh \)
\[
5gh = 5(4)(2) = 5 \cdot 8 = 40
\]
- c) \( 4fh - 2g \)
\[
4fh - 2g = 4(3)(2) - 2(4) = 24 - 8 = 16
\]

- Answers:
- a) \( 17 \)
- b) \( 40 \)
- c) \( 16 \)

4. When \( p = 5 \), \( q = -3 \), \( r = 6 \), find:
- a) \( 2p + q \)
\[
2p + q = 2(5) + (-3) = 10 - 3 = 7
\]
- b) \( 4(2r - 5) \)
\[
4(2r - 5) = 4(2(6) - 5) = 4(12 - 5) = 4(7) = 28
\]
- c) \( \frac{1}{2}pr - 2q \)
\[
\frac{1}{2}pr - 2q = \frac{1}{2}(5)(6) - 2(-3) = \frac{1}{2}(30) + 6 = 15 + 6 = 21
\]

- Answers:
- a) \( 7 \)
- b) \( 28 \)
- c) \( 21 \)

5. Spot the problems with the working out and/or answer:
- a) Find \( 3x + 4y \) when \( x = 2 \) and \( y = 3 \)
- Incorrect working:
\[
3x + 4y = 3(2) + 4(2) = 6 + 8 = 14
\]
- Correct working:
\[
3x + 4y = 3(2) + 4(3) = 6 + 12 = 18
\]
- Problem: The value of \( y \) was incorrectly used as 2 instead of 3.

- b) Find \( 3c^2 - 2cd \) when \( c = 3 \) and \( d = 2 \)
- Incorrect working:
\[
3c^2 - 2cd = 3(3)^2 - 2(3)(3) = 18 - 18 = 0
\]
- Correct working:
\[
3c^2 - 2cd = 3(3)^2 - 2(3)(2) = 3(9) - 2(6) = 27 - 12 = 15
\]
- Problem: The value of \( d \) was incorrectly used as 3 instead of 2.

---

## Final Answers

- Simplifying:
1. \( 4a \)
2. \( 12b \)
3. \( 5c + 14d \)
4. \( 2e + 3f \)
5. \( -8h + 7 \)
6. \( 4x^2 + 6x + 4 \)
7.
- a) Error: Should be \( 6a \)
- b) Error: Terms cannot be combined
- c) Error: Should be \( x^2 + 7x + 3 \)

- Expanding Brackets:
1. \( 4a + 12 \)
2. \( 6b - 30 \)
3. \( 14c + 21 \)
4. \( 10d + 15e \)
5. \( 3fg + 2f \)
6. \( 2h^2 - 8h \)
7. \( 17x + 18 \)
8. \( 5y - 8 \)

- Factorising:
1. \( 2(x + 4) \)
2. \( 3(x + 3) \)
3. \( 2(3x - 4) \)
4. \( 11(x + 8) \)
5. \( 4(2x - 3) \)
6. \( 16(4x) \)
7. \( 7x(1 + 2y) \)
8. \( 4x(3y - 4) \)
9. \( 5x(2x + 3) \)

- Substituting:
1.
- a) \( 20 \)
- b) \( 150 \)
- c) \( -15 \)
2.
- a) \( 7 \)
- b) \( 19 \)
- c) \( -12 \)
3.
- a) \( 17 \)
- b) \( 40 \)
- c) \( 16 \)
4.
- a) \( 7 \)
- b) \( 28 \)
- c) \( 21 \)
5.
- a) Problem: Incorrect use of \( y \)
- b) Problem: Incorrect use of \( d \)

---

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
&\text{Simplifying: } 4a, 12b, 5c + 14d, 2e + 3f, -8h + 7, 4x^2 + 6x + 4 \\
&\text{Expanding Brackets: } 4a + 12, 6b - 30, 14c + 21, 10d + 15e, 3fg + 2f, 2h^2 - 8h, 17x + 18, 5y - 8 \\
&\text{Factorising: } 2(x + 4), 3(x + 3), 2(3x - 4), 11(x + 8), 4(2x - 3), 16(4x), 7x(1 + 2y), 4x(3y - 4), 5x(2x + 3) \\
&\text{Substituting: } \\
&\quad 1. \text{a) } 20, \text{b) } 150, \text{c) } -15 \\
&\quad 2. \text{a) } 7, \text{b) } 19, \text{c) } -12 \\
&\quad 3. \text{a) } 17, \text{b) } 40, \text{c) } 16 \\
&\quad 4. \text{a) } 7, \text{b) } 28, \text{c) } 21 \\
&\quad 5. \text{a) Problem: Incorrect use of } y, \text{b) Problem: Incorrect use of } d
\end{aligned}
}
\]