https://r2.erweima.ai/mathbot/image/1726213056442t50behf4.jpg

Let's solve the equations step by step.

### For the first group of equations (Top section):

#### A:
1. \( x - (2 - 3) = 2 \)
   - Simplify inside the parentheses: \( x - (-1) = 2 \)
   - This becomes: \( x + 1 = 2 \)
   - Subtract 1 from both sides: \( x = 1 \)

2. \( x - (2 - 4) = 4 \)
   - Simplify inside the parentheses: \( x - (-2) = 4 \)
   - This becomes: \( x + 2 = 4 \)
   - Subtract 2 from both sides: \( x = 2 \)

3. \( x - (2 - 5) = 6 \)
   - Simplify inside the parentheses: \( x - (-3) = 6 \)
   - This becomes: \( x + 3 = 6 \)
   - Subtract 3 from both sides: \( x = 3 \)

4. \( x - (2 - 6) = 8 \)
   - Simplify inside the parentheses: \( x - (-4) = 8 \)
   - This becomes: \( x + 4 = 8 \)
   - Subtract 4 from both sides: \( x = 4 \)

#### B:
1. \( x - (-2 - 3) = 2 \)
   - Simplify inside the parentheses: \( x - (-5) = 2 \)
   - This becomes: \( x + 5 = 2 \)
   - Subtract 5 from both sides: \( x = -3 \)

2. \( x - (-2 - 4) = 4 \)
   - Simplify inside the parentheses: \( x - (-6) = 4 \)
   - This becomes: \( x + 6 = 4 \)
   - Subtract 6 from both sides: \( x = -2 \)

3. \( x - (-2 - 5) = 6 \)
   - Simplify inside the parentheses: \( x - (-7) = 6 \)
   - This becomes: \( x + 7 = 6 \)
   - Subtract 7 from both sides: \( x = -1 \)

4. \( x - (-2 - 6) = 8 \)
   - Simplify inside the parentheses: \( x - (-8) = 8 \)
   - This becomes: \( x + 8 = 8 \)
   - Subtract 8 from both sides: \( x = 0 \)

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### For the second group of equations (Bottom section):

#### A:
1. \( 10 - 5(4 - 2x) = 40 \)
   - Expand the equation: \( 10 - 5 \cdot 4 + 5 \cdot 2x = 40 \)
   - Simplifies to: \( 10 - 20 + 10x = 40 \)
   - Combine constants: \( -10 + 10x = 40 \)
   - Add 10 to both sides: \( 10x = 50 \)
   - Divide by 10: \( x = 5 \)

2. \( 20 - 5(4 - 2x) = 60 \)
   - Expand the equation: \( 20 - 5 \cdot 4 + 5 \cdot 2x = 60 \)
   - Simplifies to: \( 20 - 20 + 10x = 60 \)
   - Combine constants: \( 0 + 10x = 60 \)
   - Divide by 10: \( x = 6 \)

3. \( 30 - 5(4 - 2x) = 80 \)
   - Expand the equation: \( 30 - 5 \cdot 4 + 5 \cdot 2x = 80 \)
   - Simplifies to: \( 30 - 20 + 10x = 80 \)
   - Combine constants: \( 10 + 10x = 80 \)
   - Subtract 10 from both sides: \( 10x = 70 \)
   - Divide by 10: \( x = 7 \)

4. \( 40 - 5(4 - 2x) = 100 \)
   - Expand the equation: \( 40 - 5 \cdot 4 + 5 \cdot 2x = 100 \)
   - Simplifies to: \( 40 - 20 + 10x = 100 \)
   - Combine constants: \( 20 + 10x = 100 \)
   - Subtract 20 from both sides: \( 10x = 80 \)
   - Divide by 10: \( x = 8 \)

#### B:
1. \( -10(4 - 5x) = 5 \)
   - Expand the equation: \( -40 + 50x = 5 \)
   - Add 40 to both sides: \( 50x = 45 \)
   - Divide by 50: \( x = \frac{45}{50} = \frac{9}{10} \)

2. \( -12(1 - 5x) = 6 \)
   - Expand the equation: \( -12 + 60x = 6 \)
   - Add 12 to both sides: \( 60x = 18 \)
   - Divide by 60: \( x = \frac{18}{60} = \frac{3}{10} \)

3. \( -14(-2 - 5x) = 7 \)
   - Expand the equation: \( 28 + 70x = 7 \)
   - Subtract 28 from both sides: \( 70x = -21 \)
   - Divide by 70: \( x = \frac{-21}{70} = \frac{-3}{10} \)

4. \( -16(-5 - 5x) = 8 \)
   - Expand the equation: \( 80 + 80x = 8 \)
   - Subtract 80 from both sides: \( 80x = -72 \)
   - Divide by 80: \( x = \frac{-72}{80} = \frac{-9}{10} \)

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Would you like more details or have any specific questions? 

Here are some related questions to try:
1. Can you solve similar equations with more complex constants?
2. How would these solutions change if we altered the constants on the right-hand side?
3. What happens if we modify the coefficients of \(x\) in these equations?
4. How can we solve similar equations involving fractions instead of integers?
5. Can you derive an equation that fits a similar pattern but involves quadratic terms?

**Tip:** Always simplify inside parentheses first, and then proceed step by step to isolate the variable.

Solve for x in the given algebraic equations.

Learn how to solve linear equations for x with step-by-step solutions. This includes simplifying expressions and isolating the variable using properties of equality. Suitable for students in Grades 6-8.

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