Solving Algebraic Equations exercise - Free Printable

Let's solve each algebraic equation step by step. The goal is to isolate the variable on one side of the equation.

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1. \( 2(3 - h) - 6 = -5h \)

#### Step 1: Distribute the 2
\[ 2(3 - h) - 6 = -5h \]
\[ 6 - 2h - 6 = -5h \]

#### Step 2: Simplify
\[ 6 - 6 - 2h = -5h \]
\[ -2h = -5h \]

#### Step 3: Add \( 5h \) to both sides
\[ -2h + 5h = -5h + 5h \]
\[ 3h = 0 \]

#### Step 4: Solve for \( h \)
\[ h = 0 \]

Answer: \( h = 0 \)

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2. \( 7 + 9d = 7d + 3 \)

#### Step 1: Subtract \( 7d \) from both sides
\[ 7 + 9d - 7d = 7d + 3 - 7d \]
\[ 7 + 2d = 3 \]

#### Step 2: Subtract 7 from both sides
\[ 7 + 2d - 7 = 3 - 7 \]
\[ 2d = -4 \]

#### Step 3: Divide by 2
\[ d = \frac{-4}{2} \]
\[ d = -2 \]

Answer: \( d = -2 \)

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3. \( -2(4 + 3y) = -2(4 + y) \)

#### Step 1: Distribute the \(-2\) on both sides
\[ -2(4 + 3y) = -2(4 + y) \]
\[ -8 - 6y = -8 - 2y \]

#### Step 2: Add 8 to both sides
\[ -8 - 6y + 8 = -8 - 2y + 8 \]
\[ -6y = -2y \]

#### Step 3: Add \( 2y \) to both sides
\[ -6y + 2y = -2y + 2y \]
\[ -4y = 0 \]

#### Step 4: Solve for \( y \)
\[ y = 0 \]

Answer: \( y = 0 \)

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4. \( -7 + 4c = 7c + 6 \)

#### Step 1: Subtract \( 4c \) from both sides
\[ -7 + 4c - 4c = 7c + 6 - 4c \]
\[ -7 = 3c + 6 \]

#### Step 2: Subtract 6 from both sides
\[ -7 - 6 = 3c + 6 - 6 \]
\[ -13 = 3c \]

#### Step 3: Divide by 3
\[ c = \frac{-13}{3} \]

Answer: \( c = -\frac{13}{3} \)

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5. \( 5(1 + s) = -9s + 6 \)

#### Step 1: Distribute the 5
\[ 5(1 + s) = -9s + 6 \]
\[ 5 + 5s = -9s + 6 \]

#### Step 2: Add \( 9s \) to both sides
\[ 5 + 5s + 9s = -9s + 6 + 9s \]
\[ 5 + 14s = 6 \]

#### Step 3: Subtract 5 from both sides
\[ 5 + 14s - 5 = 6 - 5 \]
\[ 14s = 1 \]

#### Step 4: Divide by 14
\[ s = \frac{1}{14} \]

Answer: \( s = \frac{1}{14} \)

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6. \( 3 + v = 2(2v - 1) \)

#### Step 1: Distribute the 2
\[ 3 + v = 2(2v - 1) \]
\[ 3 + v = 4v - 2 \]

#### Step 2: Subtract \( v \) from both sides
\[ 3 + v - v = 4v - 2 - v \]
\[ 3 = 3v - 2 \]

#### Step 3: Add 2 to both sides
\[ 3 + 2 = 3v - 2 + 2 \]
\[ 5 = 3v \]

#### Step 4: Divide by 3
\[ v = \frac{5}{3} \]

Answer: \( v = \frac{5}{3} \)

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7. \( -2 - 4w = 7w - 8 \)

#### Step 1: Add \( 4w \) to both sides
\[ -2 - 4w + 4w = 7w - 8 + 4w \]
\[ -2 = 11w - 8 \]

#### Step 2: Add 8 to both sides
\[ -2 + 8 = 11w - 8 + 8 \]
\[ 6 = 11w \]

#### Step 3: Divide by 11
\[ w = \frac{6}{11} \]

Answer: \( w = \frac{6}{11} \)

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8. \( -6(1 - m) = 9 - 2m \)

#### Step 1: Distribute the \(-6\)
\[ -6(1 - m) = 9 - 2m \]
\[ -6 + 6m = 9 - 2m \]

#### Step 2: Add \( 2m \) to both sides
\[ -6 + 6m + 2m = 9 - 2m + 2m \]
\[ -6 + 8m = 9 \]

#### Step 3: Add 6 to both sides
\[ -6 + 8m + 6 = 9 + 6 \]
\[ 8m = 15 \]

#### Step 4: Divide by 8
\[ m = \frac{15}{8} \]

Answer: \( m = \frac{15}{8} \)

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9. \( -2q - 3 = -2(2q + 1) \)

#### Step 1: Distribute the \(-2\)
\[ -2q - 3 = -2(2q + 1) \]
\[ -2q - 3 = -4q - 2 \]

#### Step 2: Add \( 4q \) to both sides
\[ -2q - 3 + 4q = -4q - 2 + 4q \]
\[ 2q - 3 = -2 \]

#### Step 3: Add 3 to both sides
\[ 2q - 3 + 3 = -2 + 3 \]
\[ 2q = 1 \]

#### Step 4: Divide by 2
\[ q = \frac{1}{2} \]

Answer: \( q = \frac{1}{2} \)

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10. \( 6n + 7 = 2n + 5 \)

#### Step 1: Subtract \( 2n \) from both sides
\[ 6n + 7 - 2n = 2n + 5 - 2n \]
\[ 4n + 7 = 5 \]

#### Step 2: Subtract 7 from both sides
\[ 4n + 7 - 7 = 5 - 7 \]
\[ 4n = -2 \]

#### Step 3: Divide by 4
\[ n = \frac{-2}{4} \]
\[ n = -\frac{1}{2} \]

Answer: \( n = -\frac{1}{2} \)

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Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ h = 0 \\
2. & \ d = -2 \\
3. & \ y = 0 \\
4. & \ c = -\frac{13}{3} \\
5. & \ s = \frac{1}{14} \\
6. & \ v = \frac{5}{3} \\
7. & \ w = \frac{6}{11} \\
8. & \ m = \frac{15}{8} \\
9. & \ q = \frac{1}{2} \\
10. & \ n = -\frac{1}{2}
\end{aligned}
}
\]