Solving Equations Worksheets - Access Maths - Free Printable

Problem: Solving Equations with Unknowns on Both Sides

The task involves solving equations where the variable (unknown) appears on both sides of the equation. We will solve each equation step by step, starting from the Bronze level and moving to the Gold level.

---

Bronze Level

#### a) \( 8x + 8 = 2x + 62 \)
1. Subtract \( 2x \) from both sides:
\[
8x - 2x + 8 = 62
\]
\[
6x + 8 = 62
\]
2. Subtract 8 from both sides:
\[
6x = 54
\]
3. Divide by 6:
\[
x = 9
\]

#### b) \( 7x + 3 = x + 33 \)
1. Subtract \( x \) from both sides:
\[
7x - x + 3 = 33
\]
\[
6x + 3 = 33
\]
2. Subtract 3 from both sides:
\[
6x = 30
\]
3. Divide by 6:
\[
x = 5
\]

#### c) \( 6x + 5 = 3x + 17 \)
1. Subtract \( 3x \) from both sides:
\[
6x - 3x + 5 = 17
\]
\[
3x + 5 = 17
\]
2. Subtract 5 from both sides:
\[
3x = 12
\]
3. Divide by 3:
\[
x = 4
\]

#### d) \( 7x + 7 = 3x + 7 \)
1. Subtract \( 3x \) from both sides:
\[
7x - 3x + 7 = 7
\]
\[
4x + 7 = 7
\]
2. Subtract 7 from both sides:
\[
4x = 0
\]
3. Divide by 4:
\[
x = 0
\]

#### e) \( 7x + 10 = 4x + 25 \)
1. Subtract \( 4x \) from both sides:
\[
7x - 4x + 10 = 25
\]
\[
3x + 10 = 25
\]
2. Subtract 10 from both sides:
\[
3x = 15
\]
3. Divide by 3:
\[
x = 5
\]

#### f) \( 7x + 4 = 2x + 29 \)
1. Subtract \( 2x \) from both sides:
\[
7x - 2x + 4 = 29
\]
\[
5x + 4 = 29
\]
2. Subtract 4 from both sides:
\[
5x = 25
\]
3. Divide by 5:
\[
x = 5
\]

#### g) \( 5x + 10 = 4x + 14 \)
1. Subtract \( 4x \) from both sides:
\[
5x - 4x + 10 = 14
\]
\[
x + 10 = 14
\]
2. Subtract 10 from both sides:
\[
x = 4
\]

#### h) \( 8x + 1 = 4x + 29 \)
1. Subtract \( 4x \) from both sides:
\[
8x - 4x + 1 = 29
\]
\[
4x + 1 = 29
\]
2. Subtract 1 from both sides:
\[
4x = 28
\]
3. Divide by 4:
\[
x = 7
\]

---

Silver Level

#### a) \( 10(x - 8) = 3x - 66 \)
1. Distribute the 10 on the left side:
\[
10x - 80 = 3x - 66
\]
2. Subtract \( 3x \) from both sides:
\[
10x - 3x - 80 = -66
\]
\[
7x - 80 = -66
\]
3. Add 80 to both sides:
\[
7x = 14
\]
4. Divide by 7:
\[
x = 2
\]

#### b) \( 5(x + 4) = 9x - 12 \)
1. Distribute the 5 on the left side:
\[
5x + 20 = 9x - 12
\]
2. Subtract \( 5x \) from both sides:
\[
20 = 9x - 5x - 12
\]
\[
20 = 4x - 12
\]
3. Add 12 to both sides:
\[
32 = 4x
\]
4. Divide by 4:
\[
x = 8
\]

#### c) \( 7(x + 5) = 10x + 20 \)
1. Distribute the 7 on the left side:
\[
7x + 35 = 10x + 20
\]
2. Subtract \( 7x \) from both sides:
\[
35 = 10x - 7x + 20
\]
\[
35 = 3x + 20
\]
3. Subtract 20 from both sides:
\[
15 = 3x
\]
4. Divide by 3:
\[
x = 5
\]

#### d) \( 7(x - 2) = 9x - 16 \)
1. Distribute the 7 on the left side:
\[
7x - 14 = 9x - 16
\]
2. Subtract \( 7x \) from both sides:
\[
-14 = 9x - 7x - 16
\]
\[
-14 = 2x - 16
\]
3. Add 16 to both sides:
\[
2 = 2x
\]
4. Divide by 2:
\[
x = 1
\]

#### e) \( 6(x + 3) = 10x - 18 \)
1. Distribute the 6 on the left side:
\[
6x + 18 = 10x - 18
\]
2. Subtract \( 6x \) from both sides:
\[
18 = 10x - 6x - 18
\]
\[
18 = 4x - 18
\]
3. Add 18 to both sides:
\[
36 = 4x
\]
4. Divide by 4:
\[
x = 9
\]

#### f) \( 9(x + 2) = 4x + 23 \)
1. Distribute the 9 on the left side:
\[
9x + 18 = 4x + 23
\]
2. Subtract \( 4x \) from both sides:
\[
9x - 4x + 18 = 23
\]
\[
5x + 18 = 23
\]
3. Subtract 18 from both sides:
\[
5x = 5
\]
4. Divide by 5:
\[
x = 1
\]

#### g) \( 3(x - 10) = 2x - 25 \)
1. Distribute the 3 on the left side:
\[
3x - 30 = 2x - 25
\]
2. Subtract \( 2x \) from both sides:
\[
3x - 2x - 30 = -25
\]
\[
x - 30 = -25
\]
3. Add 30 to both sides:
\[
x = 5
\]

#### h) \( 10(x + 6) = 3x + 60 \)
1. Distribute the 10 on the left side:
\[
10x + 60 = 3x + 60
\]
2. Subtract \( 3x \) from both sides:
\[
10x - 3x + 60 = 60
\]
\[
7x + 60 = 60
\]
3. Subtract 60 from both sides:
\[
7x = 0
\]
4. Divide by 7:
\[
x = 0
\]

---

Gold Level

#### a) \( 2(p + 6) = 6(p - 0) \)
1. Simplify the right side (\( p - 0 = p \)):
\[
2(p + 6) = 6p
\]
2. Distribute the 2 on the left side:
\[
2p + 12 = 6p
\]
3. Subtract \( 2p \) from both sides:
\[
12 = 6p - 2p
\]
\[
12 = 4p
\]
4. Divide by 4:
\[
p = 3
\]

#### b) \( 6(3x + 6) = 4(5x - 2) \)
1. Distribute on both sides:
\[
18x + 36 = 20x - 8
\]
2. Subtract \( 18x \) from both sides:
\[
36 = 20x - 18x - 8
\]
\[
36 = 2x - 8
\]
3. Add 8 to both sides:
\[
44 = 2x
\]
4. Divide by 2:
\[
x = 22
\]

#### c) \( 5(a + 3) = 4(2a + 3) \)
1. Distribute on both sides:
\[
5a + 15 = 8a + 12
\]
2. Subtract \( 5a \) from both sides:
\[
15 = 8a - 5a + 12
\]
\[
15 = 3a + 12
\]
3. Subtract 12 from both sides:
\[
3 = 3a
\]
4. Divide by 3:
\[
a = 1
\]

#### d) \( 4(b + 3) = 2(b + 4) \)
1. Distribute on both sides:
\[
4b + 12 = 2b + 8
\]
2. Subtract \( 2b \) from both sides:
\[
4b - 2b + 12 = 8
\]
\[
2b + 12 = 8
\]
3. Subtract 12 from both sides:
\[
2b = -4
\]
4. Divide by 2:
\[
b = -2
\]

#### e) \( 3(2y + 1) = 5(y + 2) \)
1. Distribute on both sides:
\[
6y + 3 = 5y + 10
\]
2. Subtract \( 5y \) from both sides:
\[
6y - 5y + 3 = 10
\]
\[
y + 3 = 10
\]
3. Subtract 3 from both sides:
\[
y = 7
\]

#### f) \( 5(2x + 3) = 3(3x + 5) \)
1. Distribute on both sides:
\[
10x + 15 = 9x + 15
\]
2. Subtract \( 9x \) from both sides:
\[
10x - 9x + 15 = 15
\]
\[
x + 15 = 15
\]
3. Subtract 15 from both sides:
\[
x = 0
\]

#### g) \( 3(5x + 5) = 4(4x + 5) \)
1. Distribute on both sides:
\[
15x + 15 = 16x + 20
\]
2. Subtract \( 15x \) from both sides:
\[
15 = 16x - 15x + 20
\]
\[
15 = x + 20
\]
3. Subtract 20 from both sides:
\[
-5 = x
\]
\[
x = -5
\]

#### h) \( 4(t + 4) = 6(t + 3) \)
1. Distribute on both sides:
\[
4t + 16 = 6t + 18
\]
2. Subtract \( 4t \) from both sides:
\[
16 = 6t - 4t + 18
\]
\[
16 = 2t + 18
\]
3. Subtract 18 from both sides:
\[
-2 = 2t
\]
4. Divide by 2:
\[
t = -1
\]

---

Final Answers

\[
\boxed{
\begin{aligned}
&\text{Bronze: } a) 9, b) 5, c) 4, d) 0, e) 5, f) 5, g) 4, h) 7 \\
&\text{Silver: } a) 2, b) 8, c) 5, d) 1, e) 9, f) 1, g) 5, h) 0 \\
&\text{Gold: } a) 3, b) 22, c) 1, d) -2, e) 7, f) 0, g) -5, h) -1
\end{aligned}
}
\]