How To Solve 2 Step Equation Division

Solving two-step equations involving division might seem daunting at first, but with a clear understanding of the underlying principles and a systematic approach, anyone can master this skill. The key lies in isolating the variable, and this involves undoing the operations in the reverse order of the order of operations (PEMDAS/BODMAS). This article will provide a comprehensive guide on how to solve two-step equations that involve division, complete with examples and explanations to ensure clarity.

Understanding Two-Step Equations

A two-step equation is an algebraic equation that requires two operations to isolate the variable. These operations typically involve a combination of addition, subtraction, multiplication, and division. Equations involving division usually present the variable being divided by a number, and sometimes, this can be intertwined with addition or subtraction.

General Form

The general form of a two-step equation involving division can be represented as:

(x / a) + b = c

or

(x / a) - b = c

Where:

• x is the variable we need to solve for.
• a is the divisor (the number x is divided by).
• b is a constant that is either added to or subtracted from the term containing x.
• c is the constant on the other side of the equation.

The Goal

The primary goal in solving any equation is to isolate the variable on one side of the equation. This means manipulating the equation to get x = some value. To achieve this, we need to undo the operations affecting x in the reverse order.

Steps to Solve Two-Step Equations with Division

Solving two-step equations involving division requires a systematic approach. Here’s a step-by-step guide to help you navigate through the process:

Step 1: Identify the Operations

The first step is to identify the operations that are being applied to the variable. In an equation like (x / a) + b = c, the variable x is first divided by a, and then b is added to the result.

Step 2: Undo Addition or Subtraction

The second step is to undo any addition or subtraction. This is done by performing the inverse operation on both sides of the equation.

• If b is added, subtract b from both sides of the equation.

  (x / a) + b - b = c - b

  This simplifies to:

  x / a = c - b

• If b is subtracted, add b to both sides of the equation.

  (x / a) - b + b = c + b

  This simplifies to:

  x / a = c + b

Step 3: Undo Division

The third step is to undo the division. This is done by multiplying both sides of the equation by a.

• If the equation is x / a = c - b, multiply both sides by a.

  (x / a) * a = (c - b) * a

  This simplifies to:

  x = (c - b) * a

• If the equation is x / a = c + b, multiply both sides by a.

  (x / a) * a = (c + b) * a

  This simplifies to:

  x = (c + b) * a

Step 4: Simplify and Solve

The final step is to simplify the equation and solve for x. This involves performing the arithmetic operations to find the value of x.

Example Problems

Let's walk through some examples to illustrate the steps involved in solving two-step equations with division.

Example 1

Solve the equation:

(x / 3) + 5 = 9

1. Identify the operations: x is divided by 3, and 5 is added to the result.

2. Undo addition: Subtract 5 from both sides of the equation.

   (x / 3) + 5 - 5 = 9 - 5

   x / 3 = 4

3. Undo division: Multiply both sides by 3.

   (x / 3) * 3 = 4 * 3

   x = 12

4. Simplify: The value of x is 12.

   x = 12

Example 2

Solve the equation:

(x / 4) - 2 = 6

1. Identify the operations: x is divided by 4, and 2 is subtracted from the result.

2. Undo subtraction: Add 2 to both sides of the equation.

   (x / 4) - 2 + 2 = 6 + 2

   x / 4 = 8

3. Undo division: Multiply both sides by 4.

   (x / 4) * 4 = 8 * 4

   x = 32

4. Simplify: The value of x is 32.

   x = 32

Example 3

Solve the equation:

(x / -2) + 3 = 1

1. Identify the operations: x is divided by -2, and 3 is added to the result.

2. Undo addition: Subtract 3 from both sides of the equation.

   (x / -2) + 3 - 3 = 1 - 3

   x / -2 = -2

3. Undo division: Multiply both sides by -2.

   (x / -2) * -2 = -2 * -2

   x = 4

4. Simplify: The value of x is 4.

   x = 4

Example 4

Solve the equation:

(x / 5) - 4 = -1

1. Identify the operations: x is divided by 5, and 4 is subtracted from the result.

2. Undo subtraction: Add 4 to both sides of the equation.

   (x / 5) - 4 + 4 = -1 + 4

   x / 5 = 3

3. Undo division: Multiply both sides by 5.

   (x / 5) * 5 = 3 * 5

   x = 15

4. Simplify: The value of x is 15.

   x = 15

Example 5

Solve the equation:

(x / -3) - 1 = 5

1. Identify the operations: x is divided by -3, and 1 is subtracted from the result.

2. Undo subtraction: Add 1 to both sides of the equation.

   (x / -3) - 1 + 1 = 5 + 1

   x / -3 = 6

3. Undo division: Multiply both sides by -3.

   (x / -3) * -3 = 6 * -3

   x = -18

4. Simplify: The value of x is -18.

   x = -18

Advanced Scenarios and Tips

Dealing with Negative Numbers

When dealing with negative numbers, it's crucial to pay close attention to the signs. Remember the rules for multiplying and dividing negative numbers:

• A negative number multiplied by a negative number results in a positive number.
• A negative number multiplied by a positive number results in a negative number.
• A positive number divided by a negative number results in a negative number.
• A negative number divided by a negative number results in a positive number.

Fractions in the Equation

Sometimes, you might encounter equations where the constant term b or c are fractions. In such cases, it's important to perform the operations with fractions accurately.

For example, consider the equation:

(x / 2) + (1 / 3) = (5 / 6)

1. Undo addition: Subtract 1/3 from both sides of the equation.

   (x / 2) + (1 / 3) - (1 / 3) = (5 / 6) - (1 / 3)

   x / 2 = (5 / 6) - (2 / 6)

   x / 2 = 3 / 6

   x / 2 = 1 / 2

2. Undo division: Multiply both sides by 2.

   (x / 2) * 2 = (1 / 2) * 2

   x = 1

3. Simplify: The value of x is 1.

   x = 1

Complex Fractions

If the equation involves complex fractions, simplify the fractions before proceeding with the steps to solve the equation. For example:

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

First, simplify the complex fraction (x / 3) / 2. This is equivalent to x / 6.

(x / 6) + 1 = 4

Now, solve the equation as usual:

1. Undo addition: Subtract 1 from both sides.

   (x / 6) + 1 - 1 = 4 - 1

   x / 6 = 3

2. Undo division: Multiply both sides by 6.

   (x / 6) * 6 = 3 * 6

   x = 18

3. Simplify: The value of x is 18.

   x = 18

Equations with Parentheses

Sometimes, the equation might have parentheses. In such cases, apply the distributive property to remove the parentheses before solving the equation. For example:

(x / 2 + 1) * 3 = 12

First, divide both sides by 3 to remove the multiplication:

(x / 2 + 1) = 4

Now, solve the equation as usual:

1. Undo addition: Subtract 1 from both sides.

   (x / 2) + 1 - 1 = 4 - 1

   x / 2 = 3

2. Undo division: Multiply both sides by 2.

   (x / 2) * 2 = 3 * 2

   x = 6

3. Simplify: The value of x is 6.

   x = 6

Common Mistakes to Avoid

When solving two-step equations, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

• Incorrect Order of Operations: Always remember to undo the operations in the reverse order of PEMDAS/BODMAS. Failing to do so will lead to incorrect results.
• Not Applying Operations to Both Sides: Any operation you perform on one side of the equation must also be performed on the other side to maintain the equality.
• Sign Errors: Be extra cautious when dealing with negative numbers. A small mistake in the sign can lead to a completely different answer.
• Arithmetic Errors: Double-check your arithmetic calculations to avoid simple mistakes that can throw off your solution.
• Forgetting to Simplify: Always simplify the equation as much as possible before and after each step to make the problem easier to solve.

Practice Problems

To solidify your understanding, here are some practice problems to work through:

1. (x / 5) + 2 = 7
2. (x / 3) - 1 = 4
3. (x / -2) + 4 = 2
4. (x / 4) - 3 = -1
5. (x / -5) - 2 = 3
6. (x / 6) + 5 = 10
7. (x / -3) + 1 = -2
8. (x / 2) - 4 = 0
9. (x / 7) + 3 = 5
10. (x / -4) - 1 = 1

Answers:

1. x = 25
2. x = 15
3. x = 4
4. x = 8
5. x = -25
6. x = 30
7. x = 9
8. x = 8
9. x = 14
10. x = -8

Conclusion

Solving two-step equations involving division is a fundamental skill in algebra. By following the steps outlined in this guide, you can confidently solve these equations. Remember to identify the operations, undo them in the correct order, and simplify your results. With practice, you'll become proficient at solving these types of equations, and you'll be well-prepared for more advanced algebraic concepts. Always double-check your work and be mindful of common mistakes to ensure accuracy.