Algebra worksheet with equations containing variables on both sides, including examples, workout questions, and a video link.

Problem: Solve the following equations where letters appear on both sides.

#### Equations:
1. \( 4x + 1 = 2x + 7 \)
2. \( 5x + 4 = 3x + 16 \)
3. \( 2x + 8 = x + 12 \)
4. \( 7x + 1 = 2x + 46 \)
5. \( 6x - 3 = 2x + 13 \)
6. \( 9x - 10 = 7x + 24 \)
7. \( 2x + 21 = 4x + 5 \)
8. \( x + 2 = 5x - 2 \)
9. \( 6x - 9 = 4x - 1 \)
10. \( 5x + 2 = 16 - 2x \)
11. \( 3x - 1 = 23 - x \)
12. \( 6x + 8 = 38 - 4x \)
13. \( 80 - x = 8x - 1 \)
14. \( 2x + 7 = 17 - 8x \)
15. \( 15 - x = 27 - 3x \)
16. \( 12x - 20 = 15x - 38 \)
17. \( 35x + 10 = 20x + 175 \)
18. \( 14x = 2x + 60 \)

---

Solution Approach:

To solve equations with variables on both sides, follow these steps:
1. Move all terms containing the variable to one side of the equation by adding or subtracting.
2. Move all constant terms to the other side by adding or subtracting.
3. Simplify both sides.
4. Isolate the variable by dividing or multiplying as needed.
5. Verify the solution (optional but recommended).

---

Detailed Solutions:

#### (a) \( 4x + 1 = 2x + 7 \)
1. Subtract \( 2x \) from both sides:
\[
4x - 2x + 1 = 2x - 2x + 7
\]
\[
2x + 1 = 7
\]
2. Subtract 1 from both sides:
\[
2x + 1 - 1 = 7 - 1
\]
\[
2x = 6
\]
3. Divide by 2:
\[
x = 3
\]

#### (b) \( 5x + 4 = 3x + 16 \)
1. Subtract \( 3x \) from both sides:
\[
5x - 3x + 4 = 3x - 3x + 16
\]
\[
2x + 4 = 16
\]
2. Subtract 4 from both sides:
\[
2x + 4 - 4 = 16 - 4
\]
\[
2x = 12
\]
3. Divide by 2:
\[
x = 6
\]

#### (c) \( 2x + 8 = x + 12 \)
1. Subtract \( x \) from both sides:
\[
2x - x + 8 = x - x + 12
\]
\[
x + 8 = 12
\]
2. Subtract 8 from both sides:
\[
x + 8 - 8 = 12 - 8
\]
\[
x = 4
\]

#### (d) \( 7x + 1 = 2x + 46 \)
1. Subtract \( 2x \) from both sides:
\[
7x - 2x + 1 = 2x - 2x + 46
\]
\[
5x + 1 = 46
\]
2. Subtract 1 from both sides:
\[
5x + 1 - 1 = 46 - 1
\]
\[
5x = 45
\]
3. Divide by 5:
\[
x = 9
\]

#### (e) \( 6x - 3 = 2x + 13 \)
1. Subtract \( 2x \) from both sides:
\[
6x - 2x - 3 = 2x - 2x + 13
\]
\[
4x - 3 = 13
\]
2. Add 3 to both sides:
\[
4x - 3 + 3 = 13 + 3
\]
\[
4x = 16
\]
3. Divide by 4:
\[
x = 4
\]

#### (f) \( 9x - 10 = 7x + 24 \)
1. Subtract \( 7x \) from both sides:
\[
9x - 7x - 10 = 7x - 7x + 24
\]
\[
2x - 10 = 24
\]
2. Add 10 to both sides:
\[
2x - 10 + 10 = 24 + 10
\]
\[
2x = 34
\]
3. Divide by 2:
\[
x = 17
\]

#### (g) \( 2x + 21 = 4x + 5 \)
1. Subtract \( 2x \) from both sides:
\[
2x - 2x + 21 = 4x - 2x + 5
\]
\[
21 = 2x + 5
\]
2. Subtract 5 from both sides:
\[
21 - 5 = 2x + 5 - 5
\]
\[
16 = 2x
\]
3. Divide by 2:
\[
x = 8
\]

#### (h) \( x + 2 = 5x - 2 \)
1. Subtract \( x \) from both sides:
\[
x - x + 2 = 5x - x - 2
\]
\[
2 = 4x - 2
\]
2. Add 2 to both sides:
\[
2 + 2 = 4x - 2 + 2
\]
\[
4 = 4x
\]
3. Divide by 4:
\[
x = 1
\]

#### (i) \( 6x - 9 = 4x - 1 \)
1. Subtract \( 4x \) from both sides:
\[
6x - 4x - 9 = 4x - 4x - 1
\]
\[
2x - 9 = -1
\]
2. Add 9 to both sides:
\[
2x - 9 + 9 = -1 + 9
\]
\[
2x = 8
\]
3. Divide by 2:
\[
x = 4
\]

#### (j) \( 5x + 2 = 16 - 2x \)
1. Add \( 2x \) to both sides:
\[
5x + 2x + 2 = 16 - 2x + 2x
\]
\[
7x + 2 = 16
\]
2. Subtract 2 from both sides:
\[
7x + 2 - 2 = 16 - 2
\]
\[
7x = 14
\]
3. Divide by 7:
\[
x = 2
\]

#### (k) \( 3x - 1 = 23 - x \)
1. Add \( x \) to both sides:
\[
3x + x - 1 = 23 - x + x
\]
\[
4x - 1 = 23
\]
2. Add 1 to both sides:
\[
4x - 1 + 1 = 23 + 1
\]
\[
4x = 24
\]
3. Divide by 4:
\[
x = 6
\]

#### (l) \( 6x + 8 = 38 - 4x \)
1. Add \( 4x \) to both sides:
\[
6x + 4x + 8 = 38 - 4x + 4x
\]
\[
10x + 8 = 38
\]
2. Subtract 8 from both sides:
\[
10x + 8 - 8 = 38 - 8
\]
\[
10x = 30
\]
3. Divide by 10:
\[
x = 3
\]

#### (m) \( 80 - x = 8x - 1 \)
1. Add \( x \) to both sides:
\[
80 - x + x = 8x + x - 1
\]
\[
80 = 9x - 1
\]
2. Add 1 to both sides:
\[
80 + 1 = 9x - 1 + 1
\]
\[
81 = 9x
\]
3. Divide by 9:
\[
x = 9
\]

#### (n) \( 2x + 7 = 17 - 8x \)
1. Add \( 8x \) to both sides:
\[
2x + 8x + 7 = 17 - 8x + 8x
\]
\[
10x + 7 = 17
\]
2. Subtract 7 from both sides:
\[
10x + 7 - 7 = 17 - 7
\]
\[
10x = 10
\]
3. Divide by 10:
\[
x = 1
\]

#### (o) \( 15 - x = 27 - 3x \)
1. Add \( 3x \) to both sides:
\[
15 - x + 3x = 27 - 3x + 3x
\]
\[
15 + 2x = 27
\]
2. Subtract 15 from both sides:
\[
15 + 2x - 15 = 27 - 15
\]
\[
2x = 12
\]
3. Divide by 2:
\[
x = 6
\]

#### (p) \( 12x - 20 = 15x - 38 \)
1. Subtract \( 12x \) from both sides:
\[
12x - 12x - 20 = 15x - 12x - 38
\]
\[
-20 = 3x - 38
\]
2. Add 38 to both sides:
\[
-20 + 38 = 3x - 38 + 38
\]
\[
18 = 3x
\]
3. Divide by 3:
\[
x = 6
\]

#### (q) \( 35x + 10 = 20x + 175 \)
1. Subtract \( 20x \) from both sides:
\[
35x - 20x + 10 = 20x - 20x + 175
\]
\[
15x + 10 = 175
\]
2. Subtract 10 from both sides:
\[
15x + 10 - 10 = 175 - 10
\]
\[
15x = 165
\]
3. Divide by 15:
\[
x = 11
\]

#### (r) \( 14x = 2x + 60 \)
1. Subtract \( 2x \) from both sides:
\[
14x - 2x = 2x - 2x + 60
\]
\[
12x = 60
\]
2. Divide by 12:
\[
x = 5
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
&\text{(a)}~x = 3, &\text{(b)}~x = 6, &\text{(c)}~x = 4, &\text{(d)}~x = 9, \\
&\text{(e)}~x = 4, &\text{(f)}~x = 17, &\text{(g)}~x = 8, &\text{(h)}~x = 1, \\
&\text{(i)}~x = 4, &\text{(j)}~x = 2, &\text{(k)}~x = 6, &\text{(l)}~x = 3, \\
&\text{(m)}~x = 9, &\text{(n)}~x = 1, &\text{(o)}~x = 6, &\text{(p)}~x = 6, \\
&\text{(q)}~x = 11, &\text{(r)}~x = 5.
\end{aligned}
}
\]