Concept 10: Solving Equations - Free Printable

It looks like you've uploaded a worksheet titled "Solving Linear Equations: Variable on Both Sides", but the numbering appears to be inconsistent (e.g., multiple equations labeled as "1)"). I’ll solve each equation in order as they appear, assuming the correct sequence based on layout.

Let’s go through each equation step-by-step and solve it.

---

1) $ 6r + 7 = 13 + 7r $

Step 1: Move all terms with $ r $ to one side. Subtract $ 7r $ from both sides:

$$
6r - 7r + 7 = 13
\Rightarrow -r + 7 = 13
$$

Step 2: Subtract 7 from both sides:

$$
-r = 13 - 7 = 6
$$

Step 3: Multiply both sides by -1:

$$
r = -6
$$

✔ Answer: $ r = -6 $

---

2) $ 13 - 4x = 1 - x $

Step 1: Add $ 4x $ to both sides to move all $ x $-terms to one side:

$$
13 = 1 + 3x
$$

Step 2: Subtract 1 from both sides:

$$
12 = 3x
$$

Step 3: Divide by 3:

$$
x = 4
$$

✔ Answer: $ x = 4 $

---

3) $ -7x - 3x + 2 = -8x - 8 $

Step 1: Combine like terms on the left:

$$
-10x + 2 = -8x - 8
$$

Step 2: Add $ 10x $ to both sides:

$$
2 = 2x - 8
$$

Step 3: Add 8 to both sides:

$$
10 = 2x
$$

Step 4: Divide by 2:

$$
x = 5
$$

✔ Answer: $ x = 5 $

---

4) $ -8 - x = x - 4x $

Step 1: Simplify the right-hand side:

$$
-8 - x = -3x
$$

Step 2: Add $ 3x $ to both sides:

$$
-8 + 2x = 0
$$

Step 3: Add 8 to both sides:

$$
2x = 8
$$

Step 4: Divide by 2:

$$
x = 4
$$

✔ Answer: $ x = 4 $

---

5) $ -14 + 6b + 7 - 2b = 1 + 5b $

Step 1: Combine like terms on the left:

$$
(-14 + 7) + (6b - 2b) = 1 + 5b
\Rightarrow -7 + 4b = 1 + 5b
$$

Step 2: Subtract $ 4b $ from both sides:

$$
-7 = 1 + b
$$

Step 3: Subtract 1:

$$
-8 = b
$$

✔ Answer: $ b = -8 $

---

6) $ n + 2 = -14 - n $

Step 1: Add $ n $ to both sides:

$$
2n + 2 = -14
$$

Step 2: Subtract 2:

$$
2n = -16
$$

Step 3: Divide by 2:

$$
n = -8
$$

✔ Answer: $ n = -8 $

---

7) $ n - 3n = 14 - 4n $

Step 1: Simplify left side:

$$
-2n = 14 - 4n
$$

Step 2: Add $ 4n $ to both sides:

$$
2n = 14
$$

Step 3: Divide by 2:

$$
n = 7
$$

✔ Answer: $ n = 7 $

---

8) $ 7a - 3 = 3 + 6a $

Step 1: Subtract $ 6a $ from both sides:

$$
a - 3 = 3
$$

Step 2: Add 3 to both sides:

$$
a = 6
$$

✔ Answer: $ a = 6 $

---

9) $ -10 + x + 4 - 5 = 7x - 5 $

Step 1: Simplify left side:

$$
(-10 + 4 - 5) + x = 7x - 5
\Rightarrow -11 + x = 7x - 5
$$

Step 2: Subtract $ x $ from both sides:

$$
-11 = 6x - 5
$$

Step 3: Add 5:

$$
-6 = 6x
$$

Step 4: Divide by 6:

$$
x = -1
$$

✔ Answer: $ x = -1 $

---

10) $ -8n + 4(1 + 5n) = -6n - 14 $

Step 1: Distribute the 4:

$$
-8n + 4 + 20n = -6n - 14
$$

Step 2: Combine like terms on left:

$$
12n + 4 = -6n - 14
$$

Step 3: Add $ 6n $ to both sides:

$$
18n + 4 = -14
$$

Step 4: Subtract 4:

$$
18n = -18
$$

Step 5: Divide by 18:

$$
n = -1
$$

✔ Answer: $ n = -1 $

---

11) $ -6n - 20 = -2n + 4(1 - 3n) $

Step 1: Distribute the 4:

$$
-6n - 20 = -2n + 4 - 12n
$$

Step 2: Combine like terms on right:

$$
-6n - 20 = -14n + 4
$$

Step 3: Add $ 14n $ to both sides:

$$
8n - 20 = 4
$$

Step 4: Add 20:

$$
8n = 24
$$

Step 5: Divide by 8:

$$
n = 3
$$

✔ Answer: $ n = 3 $

---

12) $ 4n - 40 = 7(-2n + 2) $

Step 1: Distribute the 7:

$$
4n - 40 = -14n + 14
$$

Step 2: Add $ 14n $ to both sides:

$$
18n - 40 = 14
$$

Step 3: Add 40:

$$
18n = 54
$$

Step 4: Divide by 18:

$$
n = 3
$$

✔ Answer: $ n = 3 $

---

13) $ 7(5a - 4) - 1 = 14 - 8a $

Step 1: Distribute the 7:

$$
35a - 28 - 1 = 14 - 8a
\Rightarrow 35a - 29 = 14 - 8a
$$

Step 2: Add $ 8a $ to both sides:

$$
43a - 29 = 14
$$

Step 3: Add 29:

$$
43a = 43
$$

Step 4: Divide by 43:

$$
a = 1
$$

✔ Answer: $ a = 1 $

---

14) $ -31 - 4x = -5 - 5(1 + 5x) $

Step 1: Distribute the -5:

$$
-31 - 4x = -5 - 5 - 25x
\Rightarrow -31 - 4x = -10 - 25x
$$

Step 2: Add $ 25x $ to both sides:

$$
-31 + 21x = -10
$$

Step 3: Add 31:

$$
21x = 21
$$

Step 4: Divide by 21:

$$
x = 1
$$

✔ Answer: $ x = 1 $

---

15) $ 38 + 7k = 8(k + 4) $

Step 1: Distribute the 8:

$$
38 + 7k = 8k + 32
$$

Step 2: Subtract $ 7k $ from both sides:

$$
38 = k + 32
$$

Step 3: Subtract 32:

$$
k = 6
$$

✔ Answer: $ k = 6 $

---

## ✔ Final Answers Summary:

| Problem | Solution |
|--------|---------|
| 1) | $ r = -6 $ |
| 2) | $ x = 4 $ |
| 3) | $ x = 5 $ |
| 4) | $ x = 4 $ |
| 5) | $ b = -8 $ |
| 6) | $ n = -8 $ |
| 7) | $ n = 7 $ |
| 8) | $ a = 6 $ |
| 9) | $ x = -1 $ |
| 10) | $ n = -1 $ |
| 11) | $ n = 3 $ |
| 12) | $ n = 3 $ |
| 13) | $ a = 1 $ |
| 14) | $ x = 1 $ |
| 15) | $ k = 6 $ |

---

Let me know if you'd like these solutions explained in a different format (like video-style steps or visual breakdowns).