Linear Equations Worksheets with Answer Key - Free Printable

Let's solve each of the linear equations step by step.

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Problem 1:

\[ 8(4a - 4) - 1 = 14 - 8a \]

#### Step 1: Distribute the 8 on the left side.
\[ 8(4a - 4) = 32a - 32 \]
So the equation becomes:
\[ 32a - 32 - 1 = 14 - 8a \]

#### Step 2: Simplify the left side.
\[ 32a - 33 = 14 - 8a \]

#### Step 3: Add \( 8a \) to both sides to get all \( a \)-terms on one side.
\[ 32a + 8a - 33 = 14 \]
\[ 40a - 33 = 14 \]

#### Step 4: Add 33 to both sides to isolate the term with \( a \).
\[ 40a = 47 \]

#### Step 5: Divide both sides by 40.
\[ a = \frac{47}{40} \]

#### Final Answer:
\[ \boxed{\frac{47}{40}} \]

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Problem 2:

\[ 4(-8x + 5) = -30x - 26 \]

#### Step 1: Distribute the 4 on the left side.
\[ 4(-8x + 5) = -32x + 20 \]
So the equation becomes:
\[ -32x + 20 = -30x - 26 \]

#### Step 2: Add \( 32x \) to both sides to get all \( x \)-terms on one side.
\[ -32x + 32x + 20 = -30x + 32x - 26 \]
\[ 20 = 2x - 26 \]

#### Step 3: Add 26 to both sides to isolate the term with \( x \).
\[ 20 + 26 = 2x \]
\[ 46 = 2x \]

#### Step 4: Divide both sides by 2.
\[ x = 23 \]

#### Final Answer:
\[ \boxed{23} \]

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Problem 3:

\[ 27 = 46 + 3x - x \]

#### Step 1: Combine like terms on the right side.
\[ 3x - x = 2x \]
So the equation becomes:
\[ 27 = 46 + 2x \]

#### Step 2: Subtract 46 from both sides to isolate the term with \( x \).
\[ 27 - 46 = 2x \]
\[ -19 = 2x \]

#### Step 3: Divide both sides by 2.
\[ x = -\frac{19}{2} \]

#### Final Answer:
\[ \boxed{-\frac{19}{2}} \]

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Problem 4:

\[ 38 + 7k = 8(k + 4) \]

#### Step 1: Distribute the 8 on the right side.
\[ 8(k + 4) = 8k + 32 \]
So the equation becomes:
\[ 38 + 7k = 8k + 32 \]

#### Step 2: Subtract \( 7k \) from both sides to get all \( k \)-terms on one side.
\[ 38 + 7k - 7k = 8k - 7k + 32 \]
\[ 38 = k + 32 \]

#### Step 3: Subtract 32 from both sides to isolate \( k \).
\[ 38 - 32 = k \]
\[ k = 6 \]

#### Final Answer:
\[ \boxed{6} \]

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Problem 5:

\[ -2(4 - x) = 6(x + 4) + 4x \]

#### Step 1: Distribute on both sides.
On the left side:
\[ -2(4 - x) = -8 + 2x \]
On the right side:
\[ 6(x + 4) = 6x + 24 \]
So the equation becomes:
\[ -8 + 2x = 6x + 24 + 4x \]

#### Step 2: Combine like terms on the right side.
\[ 6x + 4x = 10x \]
So the equation becomes:
\[ -8 + 2x = 10x + 24 \]

#### Step 3: Subtract \( 2x \) from both sides to get all \( x \)-terms on one side.
\[ -8 + 2x - 2x = 10x - 2x + 24 \]
\[ -8 = 8x + 24 \]

#### Step 4: Subtract 24 from both sides to isolate the term with \( x \).
\[ -8 - 24 = 8x \]
\[ -32 = 8x \]

#### Step 5: Divide both sides by 8.
\[ x = -4 \]

#### Final Answer:
\[ \boxed{-4} \]

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Problem 6:

\[ 0.05z + 0.2 = 0.15z - 10.5 \]

#### Step 1: Subtract \( 0.05z \) from both sides to get all \( z \)-terms on one side.
\[ 0.05z - 0.05z + 0.2 = 0.15z - 0.05z - 10.5 \]
\[ 0.2 = 0.1z - 10.5 \]

#### Step 2: Add 10.5 to both sides to isolate the term with \( z \).
\[ 0.2 + 10.5 = 0.1z \]
\[ 10.7 = 0.1z \]

#### Step 3: Divide both sides by 0.1.
\[ z = \frac{10.7}{0.1} \]
\[ z = 107 \]

#### Final Answer:
\[ \boxed{107} \]

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Problem 7:

\[ 4(6 + 4x) = 0 \]

#### Step 1: Distribute the 4 on the left side.
\[ 4(6 + 4x) = 24 + 16x \]
So the equation becomes:
\[ 24 + 16x = 0 \]

#### Step 2: Subtract 24 from both sides to isolate the term with \( x \).
\[ 24 + 16x - 24 = 0 - 24 \]
\[ 16x = -24 \]

#### Step 3: Divide both sides by 16.
\[ x = \frac{-24}{16} \]
\[ x = -\frac{3}{2} \]

#### Final Answer:
\[ \boxed{-\frac{3}{2}} \]

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Problem 8:

\[ \frac{x}{8} - \frac{x}{9} = 1 \]

#### Step 1: Find a common denominator for the fractions.
The least common denominator (LCD) of 8 and 9 is 72. Rewrite each fraction with the denominator 72:
\[ \frac{x}{8} = \frac{9x}{72} \]
\[ \frac{x}{9} = \frac{8x}{72} \]
So the equation becomes:
\[ \frac{9x}{72} - \frac{8x}{72} = 1 \]

#### Step 2: Combine the fractions.
\[ \frac{9x - 8x}{72} = 1 \]
\[ \frac{x}{72} = 1 \]

#### Step 3: Multiply both sides by 72 to solve for \( x \).
\[ x = 72 \]

#### Final Answer:
\[ \boxed{72} \]

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Final Answers:

1. \( \boxed{\frac{47}{40}} \)
2. \( \boxed{23} \)
3. \( \boxed{-\frac{19}{2}} \)
4. \( \boxed{6} \)
5. \( \boxed{-4} \)
6. \( \boxed{107} \)
7. \( \boxed{-\frac{3}{2}} \)
8. \( \boxed{72} \)