Variable on Both Sides - Equations worksheet - Free Printable

Let's solve each equation step by step. These are linear equations with variables on both sides. The goal is to isolate the variable on one side of the equation.

---

1) $-3 + 5a = 4a - 10$

Step 1: Subtract $4a$ from both sides to get all variables on one side:
$$
-3 + 5a - 4a = -10
\Rightarrow -3 + a = -10
$$

Step 2: Add 3 to both sides:
$$
a = -10 + 3 = -7
$$

✔ Answer: $a = -7$

---

2) $-2k = k - 7 - 8$

Step 1: Simplify the right-hand side:
$$
-2k = k - 15
$$

Step 2: Subtract $k$ from both sides:
$$
-2k - k = -15 \Rightarrow -3k = -15
$$

Step 3: Divide both sides by $-3$:
$$
k = \frac{-15}{-3} = 5
$$

✔ Answer: $k = 5$

---

3) $-16 + 3n = -8 - 5n$

Step 1: Add $5n$ to both sides:
$$
-16 + 3n + 5n = -8 \Rightarrow -16 + 8n = -8
$$

Step 2: Add 16 to both sides:
$$
8n = -8 + 16 = 8
$$

Step 3: Divide by 8:
$$
n = \frac{8}{8} = 1
$$

✔ Answer: $n = 1$

---

4) $4n - 1 = 6n + 8 - 8n + 15$

Step 1: Simplify the right-hand side:
$$
6n - 8n = -2n, \quad 8 + 15 = 23
\Rightarrow 4n - 1 = -2n + 23
$$

Step 2: Add $2n$ to both sides:
$$
4n + 2n - 1 = 23 \Rightarrow 6n - 1 = 23
$$

Step 3: Add 1 to both sides:
$$
6n = 24
$$

Step 4: Divide by 6:
$$
n = \frac{24}{6} = 4
$$

✔ Answer: $n = 4$

---

5) $1 - 3x = 3x + 1$

Step 1: Subtract $3x$ from both sides:
$$
1 - 3x - 3x = 1 \Rightarrow 1 - 6x = 1
$$

Step 2: Subtract 1 from both sides:
$$
-6x = 0
$$

Step 3: Divide by $-6$:
$$
x = 0
$$

✔ Answer: $x = 0$

---

6) $4r + 8 + 5 = -15 - 3r$

Step 1: Simplify both sides:
Left: $4r + 13$, Right: $-15 - 3r$

So:
$$
4r + 13 = -15 - 3r
$$

Step 2: Add $3r$ to both sides:
$$
4r + 3r + 13 = -15 \Rightarrow 7r + 13 = -15
$$

Step 3: Subtract 13:
$$
7r = -15 - 13 = -28
$$

Step 4: Divide by 7:
$$
r = \frac{-28}{7} = -4
$$

✔ Answer: $r = -4$

---

7) $-12 - 4b = 4 - 2b$

Step 1: Add $2b$ to both sides:
$$
-12 - 4b + 2b = 4 \Rightarrow -12 - 2b = 4
$$

Step 2: Add 12 to both sides:
$$
-2b = 4 + 12 = 16
$$

Step 3: Divide by $-2$:
$$
b = \frac{16}{-2} = -8
$$

✔ Answer: $b = -8$

---

8) $3n - 15 = 7n + n$

Step 1: Simplify the right-hand side:
$$
7n + n = 8n
\Rightarrow 3n - 15 = 8n
$$

Step 2: Subtract $3n$ from both sides:
$$
-15 = 8n - 3n \Rightarrow -15 = 5n
$$

Step 3: Divide by 5:
$$
n = \frac{-15}{5} = -3
$$

✔ Answer: $n = -3$

---

✔ Final Answers:

1) $a = -7$
2) $k = 5$
3) $n = 1$
4) $n = 4$
5) $x = 0$
6) $r = -4$
7) $b = -8$
8) $n = -3$

Let me know if you'd like these checked or explained further!