1-2 Additional Practice Solving Linear Equations Answer Key

1-2 Additional Practice Solving Linear Equations: Answer Key & Detailed Explanations

Solving linear equations is a fundamental skill in algebra. Mastering this skill is crucial for success in more advanced mathematical concepts. This article provides an answer key and detailed explanations for additional practice problems focused on solving linear equations. We'll cover a range of complexities, from simple one-step equations to more challenging multi-step equations involving fractions, decimals, and the distributive property. Understanding these solutions will solidify your understanding and build confidence in tackling similar problems.

Understanding Linear Equations

Before we dive into the answer key, let's briefly review the basics of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the power of one. The general form of a linear equation in one variable is:

ax + b = c

Where:

• a, b, and c are constants (numbers)
• x is the variable we are trying to solve for.

The goal when solving a linear equation is to isolate the variable (x) on one side of the equation, leaving the solution on the other side. We achieve this by applying inverse operations, maintaining balance throughout the process.

Additional Practice Problems & Answer Key

Below are several practice problems categorized by difficulty level, followed by a comprehensive answer key with step-by-step explanations.

Level 1: One-Step Equations

1. x + 5 = 12
2. x - 7 = 3
3. 3x = 18
4. x/4 = 6

Level 2: Two-Step Equations

1. 2x + 3 = 9
2. 5x - 8 = 17
3. x/2 - 5 = 1
4. 3x + 7 = 19

Level 3: Multi-Step Equations with Fractions and Decimals

1. (1/2)x + 3 = 7
2. 0.5x - 2 = 8
3. (2/3)x + 1/4 = 5/4
4. 1.2x + 3.5 = 8.9

Level 4: Equations with the Distributive Property

1. 2(x + 3) = 10
2. 3(x - 5) + 4 = 10
3. -4(2x + 1) = 12
4. 5(x + 2) - 3x = 12

Level 5: More Complex Multi-Step Equations

1. 2(3x - 4) + 5x = 11
2. 3(x + 2) - 2(x - 1) = 8
3. 0.25(x + 4) - 0.5x = 1
4. (1/3)(x - 6) + (2/3)x = 4

Answer Key with Detailed Explanations

Level 1: One-Step Equations

1. x + 5 = 12

   • Subtract 5 from both sides: x = 7

2. x - 7 = 3

   • Add 7 to both sides: x = 10

3. 3x = 18

   • Divide both sides by 3: x = 6

4. x/4 = 6

   • Multiply both sides by 4: x = 24

Level 2: Two-Step Equations

1. 2x + 3 = 9

   • Subtract 3 from both sides: 2x = 6
   • Divide both sides by 2: x = 3

2. 5x - 8 = 17

   • Add 8 to both sides: 5x = 25
   • Divide both sides by 5: x = 5

3. x/2 - 5 = 1

   • Add 5 to both sides: x/2 = 6
   • Multiply both sides by 2: x = 12

4. 3x + 7 = 19

   • Subtract 7 from both sides: 3x = 12
   • Divide both sides by 3: x = 4

Level 3: Multi-Step Equations with Fractions and Decimals

1. (1/2)x + 3 = 7

   • Subtract 3 from both sides: (1/2)x = 4
   • Multiply both sides by 2: x = 8

2. 0.5x - 2 = 8

   • Add 2 to both sides: 0.5x = 10
   • Divide both sides by 0.5: x = 20

3. (2/3)x + 1/4 = 5/4

   • Subtract 1/4 from both sides: (2/3)x = 1
   • Multiply both sides by 3/2: x = 3/2 or 1.5

4. 1.2x + 3.5 = 8.9

   • Subtract 3.5 from both sides: 1.2x = 5.4
   • Divide both sides by 1.2: x = 4.5

Level 4: Equations with the Distributive Property

1. 2(x + 3) = 10

   • Distribute the 2: 2x + 6 = 10
   • Subtract 6 from both sides: 2x = 4
   • Divide both sides by 2: x = 2

2. 3(x - 5) + 4 = 10

   • Distribute the 3: 3x - 15 + 4 = 10
   • Combine like terms: 3x - 11 = 10
   • Add 11 to both sides: 3x = 21
   • Divide both sides by 3: x = 7

3. -4(2x + 1) = 12

   • Distribute the -4: -8x - 4 = 12
   • Add 4 to both sides: -8x = 16
   • Divide both sides by -8: x = -2

4. 5(x + 2) - 3x = 12

   • Distribute the 5: 5x + 10 - 3x = 12
   • Combine like terms: 2x + 10 = 12
   • Subtract 10 from both sides: 2x = 2
   • Divide both sides by 2: x = 1

Level 5: More Complex Multi-Step Equations

1. 2(3x - 4) + 5x = 11

   • Distribute the 2: 6x - 8 + 5x = 11
   • Combine like terms: 11x - 8 = 11
   • Add 8 to both sides: 11x = 19
   • Divide both sides by 11: x = 19/11

2. 3(x + 2) - 2(x - 1) = 8

   • Distribute the 3 and -2: 3x + 6 - 2x + 2 = 8
   • Combine like terms: x + 8 = 8
   • Subtract 8 from both sides: x = 0

3. 0.25(x + 4) - 0.5x = 1

   • Distribute the 0.25: 0.25x + 1 - 0.5x = 1
   • Combine like terms: -0.25x + 1 = 1
   • Subtract 1 from both sides: -0.25x = 0
   • Divide both sides by -0.25: x = 0

4. (1/3)(x - 6) + (2/3)x = 4

   • Distribute the 1/3: (1/3)x - 2 + (2/3)x = 4
   • Combine like terms: x - 2 = 4
   • Add 2 to both sides: x = 6

This detailed answer key provides a comprehensive guide to solving various types of linear equations. Remember to practice regularly and work through problems step-by-step to build a solid understanding. Consistent practice is key to mastering this important algebraic skill. Good luck!