Mastering Distributive Property: A Step-by-Step Guide to Solving Equations
Understanding and applying the distributive property is a fundamental skill in algebra. It allows us to simplify expressions and solve equations that involve parentheses. This article provides a comprehensive, step-by-step guide to using the distributive property to solve equations, complete with examples and explanations.
## What is the Distributive Property?
The distributive property states that for any numbers a, b, and c:
* **a(b + c) = ab + ac**
* **a(b – c) = ab – ac**
In simpler terms, the distributive property allows you to multiply a single term by each term inside a set of parentheses. This effectively removes the parentheses and simplifies the expression, making it easier to solve.
## Why is the Distributive Property Important?
The distributive property is crucial because it bridges the gap between complex expressions and simpler ones. Without it, we would be unable to solve many algebraic equations, especially those involving variables inside parentheses. It’s a foundational concept that builds upon arithmetic skills and paves the way for more advanced algebraic techniques.
## Step-by-Step Guide: Solving Equations Using the Distributive Property
Here’s a detailed breakdown of how to solve equations using the distributive property. Each step will be illustrated with examples.
**Step 1: Identify the Term Outside the Parentheses**
The first step is to locate the term directly outside the parentheses. This is the term that will be distributed to each term inside the parentheses.
**Example 1:**
`2(x + 3) = 10`
In this equation, the term outside the parentheses is `2`.
**Example 2:**
`-3(2y – 5) = 9`
In this equation, the term outside the parentheses is `-3` (pay close attention to the sign).
**Step 2: Distribute the Term**
Multiply the term outside the parentheses by each term inside the parentheses. Remember to pay attention to the signs (positive or negative) when multiplying.
**Example 1 (Continuing):**
`2(x + 3) = 10`
Distribute the `2`:
`2 * x + 2 * 3 = 10`
This simplifies to:
`2x + 6 = 10`
**Example 2 (Continuing):**
`-3(2y – 5) = 9`
Distribute the `-3`:
`(-3) * (2y) + (-3) * (-5) = 9`
This simplifies to:
`-6y + 15 = 9`
**Important Note:** A common mistake is to only multiply by the first term inside the parenthesis. Ensure you multiply by *every* term within the parentheses.
**Step 3: Simplify the Equation**
After distributing, you’ll have a simpler equation without parentheses. Combine any like terms on each side of the equation.
**Example 1 (Continuing):**
We already have `2x + 6 = 10`. There are no like terms to combine on either side.
**Example 2 (Continuing):**
We have `-6y + 15 = 9`. Again, there are no like terms to combine on either side.
**Step 4: Isolate the Variable Term**
Use inverse operations to isolate the variable term. This typically involves adding or subtracting a constant from both sides of the equation.
**Example 1 (Continuing):**
`2x + 6 = 10`
Subtract `6` from both sides:
`2x + 6 – 6 = 10 – 6`
`2x = 4`
**Example 2 (Continuing):**
`-6y + 15 = 9`
Subtract `15` from both sides:
`-6y + 15 – 15 = 9 – 15`
`-6y = -6`
**Step 5: Solve for the Variable**
Finally, divide both sides of the equation by the coefficient of the variable to solve for the variable.
**Example 1 (Continuing):**
`2x = 4`
Divide both sides by `2`:
`2x / 2 = 4 / 2`
`x = 2`
**Example 2 (Continuing):**
`-6y = -6`
Divide both sides by `-6`:
`-6y / -6 = -6 / -6`
`y = 1`
**Step 6: Check Your Solution (Optional but Recommended)**
To ensure your solution is correct, substitute the value you found for the variable back into the original equation. If both sides of the equation are equal, your solution is correct.
**Example 1 (Continuing):**
Original equation: `2(x + 3) = 10`
Substitute `x = 2`:
`2(2 + 3) = 10`
`2(5) = 10`
`10 = 10` (This is true, so `x = 2` is correct)
**Example 2 (Continuing):**
Original equation: `-3(2y – 5) = 9`
Substitute `y = 1`:
`-3(2(1) – 5) = 9`
`-3(2 – 5) = 9`
`-3(-3) = 9`
`9 = 9` (This is true, so `y = 1` is correct)
## More Complex Examples
Let’s tackle some more challenging equations that require multiple steps using the distributive property.
**Example 3:**
`4(a – 2) + 3a = 15`
1. **Distribute:**
`4a – 8 + 3a = 15`
2. **Combine Like Terms:**
`7a – 8 = 15`
3. **Isolate the Variable Term:**
`7a = 23` (Add 8 to both sides)
4. **Solve for the Variable:**
`a = 23/7`
5. **Check (Optional):**
`4((23/7) – 2) + 3(23/7) = 15`
`4((23/7) – (14/7)) + (69/7) = 15`
`4(9/7) + (69/7) = 15`
`(36/7) + (69/7) = 15`
`(105/7) = 15`
`15 = 15` (Correct)
**Example 4:**
`-2(3b + 1) – 5b = 8`
1. **Distribute:**
`-6b – 2 – 5b = 8`
2. **Combine Like Terms:**
`-11b – 2 = 8`
3. **Isolate the Variable Term:**
`-11b = 10` (Add 2 to both sides)
4. **Solve for the Variable:**
`b = -10/11`
5. **Check (Optional):**
`-2(3*(-10/11) + 1) -5*(-10/11) = 8`
`-2((-30/11) + (11/11)) + (50/11) = 8`
`-2((-19/11)) + (50/11) = 8`
`(38/11) + (50/11) = 8`
`(88/11) = 8`
`8=8` (Correct)
**Example 5:**
`5(2x – 3) = 3(x + 4)`
This example involves distribution on both sides of the equation.
1. **Distribute on both sides:**
`10x – 15 = 3x + 12`
2. **Isolate the variable term on one side:**
Subtract `3x` from both sides:
`10x – 3x – 15 = 3x – 3x + 12`
`7x – 15 = 12`
3. **Isolate the variable term:**
Add 15 to both sides:
`7x – 15 + 15 = 12 + 15`
`7x = 27`
4. **Solve for x:**
`x = 27/7`
5. **Check (Optional):**
`5(2(27/7) – 3) = 3((27/7) + 4)`
`5((54/7) – (21/7)) = 3((27/7) + (28/7))`
`5(33/7) = 3(55/7)`
`165/7 = 165/7` (Correct)
## Common Mistakes to Avoid
* **Forgetting to Distribute to All Terms:** Ensure you multiply the term outside the parentheses by every term inside the parentheses.
* **Incorrect Sign Usage:** Pay close attention to the signs (positive or negative) when multiplying. A negative times a negative is positive, and a negative times a positive is negative.
* **Combining Non-Like Terms:** Only combine terms that have the same variable and exponent (or are constants).
* **Incorrect Order of Operations:** Remember to follow the order of operations (PEMDAS/BODMAS). Distribute before combining like terms or isolating the variable.
## Tips for Success
* **Practice Regularly:** The more you practice, the more comfortable you’ll become with the distributive property.
* **Show Your Work:** Writing out each step helps prevent errors and makes it easier to identify mistakes.
* **Check Your Answers:** Always check your answers by substituting your solution back into the original equation.
* **Use Visual Aids:** If you’re struggling, try using visual aids like diagrams or color-coding to help you keep track of the terms and signs.
* **Break Down Complex Problems:** Break down complex problems into smaller, more manageable steps.
## Real-World Applications
The distributive property isn’t just a theoretical concept; it has practical applications in various real-world scenarios.
* **Calculating Costs:** Imagine you’re buying 3 identical gift baskets. Each basket contains a teddy bear costing $10 and a box of chocolates costing $5. The total cost can be calculated as 3 * ($10 + $5), which simplifies to 3 * $15 = $45. Using the distributive property, you can also calculate it as (3 * $10) + (3 * $5) = $30 + $15 = $45.
* **Scaling Recipes:** If you need to double a recipe, you’re essentially distributing the factor of 2 to each ingredient. For example, if a recipe calls for 1 cup of flour and 1/2 cup of sugar, doubling the recipe would require 2 * (1 cup flour + 1/2 cup sugar), which equals 2 cups of flour and 1 cup of sugar.
* **Calculating Areas:** The distributive property can be used to calculate the area of complex shapes by breaking them down into simpler shapes. For example, the area of a rectangle with a length of (x + 3) and a width of 5 can be calculated as 5 * (x + 3) = 5x + 15.
## Conclusion
The distributive property is an essential tool for simplifying expressions and solving algebraic equations. By following the step-by-step guide outlined in this article, practicing regularly, and avoiding common mistakes, you can master this fundamental concept and confidently tackle more advanced algebraic problems. Remember to check your solutions and apply the distributive property in real-world scenarios to solidify your understanding. Happy solving!
