Two-step Equations: Whole Numbers Answer Key

Two-Step Equations: Whole Number Answer Key – A Comprehensive Guide

Solving two-step equations is a fundamental skill in algebra. This guide provides a comprehensive explanation of how to solve them, focusing on equations with whole number solutions. We'll break down the process step-by-step, provide numerous examples, and offer strategies to help you master this crucial mathematical concept. We'll also delve into common mistakes to avoid and offer tips for success.

Understanding Two-Step Equations

A two-step equation is an algebraic equation that requires two steps to solve for the unknown variable (usually represented by 'x' or another letter). These equations typically involve addition, subtraction, multiplication, and/or division. The general form of a two-step equation is:

ax + b = c

Where:

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

The goal is to isolate 'x' on one side of the equation to find its value.

The Two-Step Process: A Step-by-Step Guide

Solving two-step equations involves applying the order of operations in reverse (PEMDAS/BODMAS) – we undo addition/subtraction first, then multiplication/division.

Step 1: Undo Addition or Subtraction

Look at the term added to or subtracted from the term containing 'x' (the 'b' in our general form). Perform the inverse operation on both sides of the equation. If 'b' is added to 'ax', subtract 'b' from both sides. If 'b' is subtracted from 'ax', add 'b' to both sides.

Step 2: Undo Multiplication or Division

After completing Step 1, you'll have an equation of the form 'ax = d' (where 'd' is the result from Step 1). Now, isolate 'x' by performing the inverse operation of the coefficient 'a'. If 'a' multiplies 'x', divide both sides by 'a'. If 'a' divides 'x', multiply both sides by 'a'.

Important Note: Remember the golden rule of algebra: Whatever you do to one side of the equation, you must do to the other side to maintain balance.

Examples: Solving Two-Step Equations with Whole Numbers

Let's work through several examples to solidify your understanding.

Example 1:

2x + 5 = 11

Step 1: Subtract 5 from both sides:

2x + 5 - 5 = 11 - 5

2x = 6

Step 2: Divide both sides by 2:

2x / 2 = 6 / 2

x = 3

Therefore, the solution is x = 3.

Example 2:

3x - 7 = 8

Step 1: Add 7 to both sides:

3x - 7 + 7 = 8 + 7

3x = 15

Step 2: Divide both sides by 3:

3x / 3 = 15 / 3

x = 5

Therefore, the solution is x = 5.

Example 3:

x/4 + 2 = 6

Step 1: Subtract 2 from both sides:

x/4 + 2 - 2 = 6 - 2

x/4 = 4

Step 2: Multiply both sides by 4:

(x/4) * 4 = 4 * 4

x = 16

Therefore, the solution is x = 16.

Example 4:

(x/5) - 3 = 2

Step 1: Add 3 to both sides:

(x/5) - 3 + 3 = 2 + 3

x/5 = 5

Step 2: Multiply both sides by 5:

(x/5) * 5 = 5 * 5

x = 25

Therefore, the solution is x = 25.

Example 5 (with a negative coefficient):

-2x + 1 = 9

Step 1: Subtract 1 from both sides:

-2x + 1 - 1 = 9 - 1

-2x = 8

Step 2: Divide both sides by -2:

-2x / -2 = 8 / -2

x = -4

Therefore, the solution is x = -4. Note that even though the instructions specified whole numbers, negative whole numbers are included.

Common Mistakes to Avoid

Several common errors can hinder your ability to solve two-step equations accurately. Here are some to watch out for:

• Incorrect order of operations: Remember to undo addition/subtraction before multiplication/division.
• Forgetting to perform the same operation on both sides: Always maintain the balance of the equation.
• Errors with negative numbers: Pay close attention to signs when adding, subtracting, multiplying, and dividing negative numbers.
• Division by zero: This is undefined. If you encounter a situation where you're about to divide by zero, re-examine your steps.

Practice Problems: Test Your Skills

To solidify your understanding, try solving these practice problems:

1. 4x + 3 = 19
2. 5x - 8 = 17
3. x/2 + 5 = 9
4. 3x + 1 = 10
5. 6x - 2 = 22
6. x/7 + 4 = 6
7. -3x + 5 = -4
8. -x/2 + 1 = 3
9. 2x - 1 = 7
10. 4x + 5 = 25

Answer Key: Check Your Work

Here are the answers to the practice problems. Remember to check your work carefully!

1. x = 4
2. x = 5
3. x = 8
4. x = 3
5. x = 4
6. x = 14
7. x = 3
8. x = -4
9. x = 4
10. x = 5

Advanced Concepts & Further Exploration

While this guide focuses on two-step equations with whole number solutions, the principles can be extended to equations involving fractions, decimals, and variables on both sides. As you progress in your algebra studies, you'll encounter more complex equations that build upon these fundamental skills.

Conclusion: Mastering Two-Step Equations

Solving two-step equations is a crucial building block in algebra. By understanding the process, practicing regularly, and avoiding common mistakes, you'll develop the confidence and proficiency needed to tackle more advanced mathematical concepts. Remember that consistent practice is key to mastering any mathematical skill. Continue to practice solving various equations to build your understanding and speed. With dedication and effort, you'll master the art of solving two-step equations.