Linear Equations Worksheet from Math Monks, designed for solving algebraic equations.

Let's solve each linear equation step by step and explain the process.

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1. $ 8(4a - 4) - 1 = 14 - 8a $

Step 1: Expand the left side:
$$
8 \cdot 4a = 32a,\quad 8 \cdot (-4) = -32
$$
So:
$$
32a - 32 - 1 = 14 - 8a
\Rightarrow 32a - 33 = 14 - 8a
$$

Step 2: Add $8a$ to both sides:
$$
32a + 8a - 33 = 14
\Rightarrow 40a - 33 = 14
$$

Step 3: Add 33 to both sides:
$$
40a = 14 + 33 = 47
$$

Step 4: Divide by 40:
$$
a = \frac{47}{40}
$$

✔ Answer: $ a = \frac{47}{40} $

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2. $ 4(-8x + 5) = -30x - 26 $

Step 1: Expand the left side:
$$
4 \cdot (-8x) = -32x,\quad 4 \cdot 5 = 20
\Rightarrow -32x + 20 = -30x - 26
$$

Step 2: Add $32x$ to both sides:
$$
20 = 2x - 26
$$

Step 3: Add 26 to both sides:
$$
46 = 2x
$$

Step 4: Divide by 2:
$$
x = 23
$$

✔ Answer: $ x = 23 $

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3. $ 27 = 46 + 3x - x $

Step 1: Simplify the right side:
$$
3x - x = 2x
\Rightarrow 27 = 46 + 2x
$$

Step 2: Subtract 46 from both sides:
$$
27 - 46 = 2x
\Rightarrow -19 = 2x
$$

Step 3: Divide by 2:
$$
x = -\frac{19}{2}
$$

✔ Answer: $ x = -\frac{19}{2} $

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

Step 1: Expand the right side:
$$
8k + 32
\Rightarrow 38 + 7k = 8k + 32
$$

Step 2: Subtract $7k$ from both sides:
$$
38 = k + 32
$$

Step 3: Subtract 32:
$$
k = 6
$$

✔ Answer: $ k = 6 $

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5. $ -2(4 - x) = 6(x + 4) + 4x $

Step 1: Expand both sides:

Left: $-2 \cdot 4 = -8$, $-2 \cdot (-x) = 2x$
$$
-8 + 2x
$$

Right: $6x + 24 + 4x = 10x + 24$

Now:
$$
-8 + 2x = 10x + 24
$$

Step 2: Subtract $2x$ from both sides:
$$
-8 = 8x + 24
$$

Step 3: Subtract 24:
$$
-32 = 8x
$$

Step 4: Divide by 8:
$$
x = -4
$$

✔ Answer: $ x = -4 $

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6. $ 0.05z + 0.2 = 0.15z - 10.5 $

Step 1: Move all terms with $z$ to one side, constants to the other.

Subtract $0.05z$ from both sides:
$$
0.2 = 0.10z - 10.5
$$

Step 2: Add 10.5 to both sides:
$$
0.2 + 10.5 = 0.10z
\Rightarrow 10.7 = 0.10z
$$

Step 3: Divide by 0.10:
$$
z = \frac{10.7}{0.10} = 107
$$

✔ Answer: $ z = 107 $

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7. $ 4(6 + 4x) = 0 $

Step 1: Expand:
$$
24 + 16x = 0
$$

Step 2: Subtract 24:
$$
16x = -24
$$

Step 3: Divide by 16:
$$
x = -\frac{24}{16} = -\frac{3}{2}
$$

✔ Answer: $ x = -\frac{3}{2} $

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8. $ \frac{x}{8} - \frac{x}{9} = 1 $

Step 1: Find common denominator: LCM of 8 and 9 is 72.

Rewrite:
$$
\frac{9x}{72} - \frac{8x}{72} = 1
\Rightarrow \frac{x}{72} = 1
$$

Step 2: Multiply both sides by 72:
$$
x = 72
$$

✔ Answer: $ x = 72 $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| ① | $ a = \frac{47}{40} $ |
| ② | $ x = 23 $ |
| ③ | $ x = -\frac{19}{2} $ |
| ④ | $ k = 6 $ |
| ⑤ | $ x = -4 $ |
| ⑥ | $ z = 107 $ |
| ⑦ | $ x = -\frac{3}{2} $ |
| ⑧ | $ x = 72 $ |

Let me know if you'd like this in a printable format or need explanations for any specific steps!