1 First Step Given Distributive Property correct as 7 x 2 7x...

← Back to Questions

### Correct Answer (Justifications for Each Step): 1. **First Step (Given)**: Distributive Property (correct, as \(-7(x - 2) = -7x + 14\) by \(a(b - c) = ab - ac\)). 2. **Second Step (\(2x + 5 = -7x + 14 \to 5 = -9x + 14\))**: **Subtraction Property of Equality** (subtract \(2x\) from both sides: \(2x + 5 - 2x = -7x + 14 - 2x\)). 3. **Third Step (Given)**: Subtraction Property of Equality (subtract 14 from both sides: \(5 - 14 = -9x + 14 - 14\)). 4. **Fourth Step (\(-9 = -9x \to 1 = x\))**: **Division Property of Equality** (divide both sides by \(-9\): \(\frac{-9}{-9} = \frac{-9x}{-9}\)). 5. **Fifth Step (\(1 = x \to x = 1\))**: **Symmetric Property of Equality** (if \(a = b\), then \(b = a\); so \(1 = x\) implies \(x = 1\)). ### Detailed Steps for Solving the Equation: 1. **Original Equation**: \(2x + 5 = -7(x - 2)\). 2. **Distributive Property** (Expand \(-7(x - 2)\)): \(-7(x - 2) = -7x + (-7)(-2) = -7x + 14\), so \(2x + 5 = -7x + 14\). 3. **Subtraction Property of Equality** (Subtract \(2x\) from both sides): \(2x + 5 - 2x = -7x + 14 - 2x\) Simplify: \(5 = -9x + 14\). 4. **Subtraction Property of Equality** (Subtract 14 from both sides): \(5 - 14 = -9x + 14 - 14\) Simplify: \(-9 = -9x\). 5. **Division Property of Equality** (Divide both sides by \(-9\)): \(\frac{-9}{-9} = \frac{-9x}{-9}\) Simplify: \(1 = x\). 6. **Symmetric Property of Equality** (Rewrite \(1 = x\) as \(x = 1\)): Since \(1 = x\), by the symmetric property, \(x = 1\). ### Relevant Knowledge Points: 1. **Distributive Property**: For real numbers \(a\), \(b\), \(c\), \(a(b + c) = ab + ac\) (or \(a(b - c) = ab - ac\)). Used to expand expressions (e.g., \(-7(x - 2) = -7x + 14\)). 2. **Subtraction Property of Equality**: If \(a = b\), then \(a - c = b - c\) for any real number \(c\). Used to isolate terms (e.g., subtracting \(2x\) or \(14\) from both sides). 3. **Division Property of Equality**: If \(a = b\) and \(c \neq 0\), then \(\frac{a}{c} = \frac{b}{c}\). Used to solve for the variable (e.g., dividing by \(-9\) to isolate \(x\)). 4. **Symmetric Property of Equality**: If \(a = b\), then \(b = a\). Used to rewrite equations in standard form (e.g., \(x = 1\) instead of \(1 = x\)). ### Explanations of Knowledge Points: - **Distributive Property**: Breaks down multiplication over addition/subtraction (e.g., \(3(x + 2) = 3x + 6\)). Critical for simplifying algebraic expressions. - **Subtraction Property of Equality**: Maintains balance by subtracting the same value from both sides (e.g., if \(x + 5 = 8\), subtract 5: \(x = 3\)). - **Division Property of Equality**: Maintains balance by dividing both sides by a non-zero number (e.g., if \(2x = 6\), divide by 2: \(x = 3\)). - **Symmetric Property of Equality**: Reflects equality (e.g., if \(5 = x\), then \(x = 5\)). Ensures solutions are written clearly. These properties ensure each step is valid, leading to the solution \(x = 1\).

Loading solution...