Use The Distributive Property To Solve The Equation 2/3 (6a + 9) = 20.4
When working with algebraic expressions, understanding how to apply the distributive property is essential for solving equations efficiently. In this article, we'll explore how to use the distributive property to solve the equation 2/3 (6a + 9) = 20.4. By breaking down each step, you'll gain confidence in tackling similar algebraic problems using this fundamental property.
Understanding the Distributive Property
Before diving into solving the equation, it's important to review what the distributive property entails and how it applies to algebra.
What Is The Distributive Property?
The distributive property states that for all numbers a, b, and c:
- a(b + c) = ab + ac
- It allows us to distribute a multiplication across addition or subtraction inside parentheses.
This property is useful because it simplifies expressions and helps isolate variables when solving equations.
Why Use The Distributive Property?
Applying the distributive property:
- Simplifies complex expressions
- Prepares equations for further operations
- Clarifies the steps needed to isolate the variable
In the context of the equation 2/3 (6a + 9) = 20.4, distributing helps eliminate parentheses and makes the equation easier to solve.
Step-by-Step Solution of 2/3 (6a + 9) = 20.4
Let's now walk through solving the equation using the distributive property.
Step 1: Apply the Distributive Property
Start by distributing 2/3 to both terms inside the parentheses:
- Multiply 2/3 by 6a:
\[
\frac{2}{3} \times 6a = \frac{2 \times 6a}{3} = \frac{12a}{3} = 4a
\]
- Multiply 2/3 by 9:
\[
\frac{2}{3} \times 9 = \frac{2 \times 9}{3} = \frac{18}{3} = 6
\]
Now, rewrite the original equation with the distributed terms:
\[
4a + 6 = 20.4
\]
Step 2: Isolate the Variable Term
Subtract 6 from both sides to get all variable terms on one side:
\[
4a + 6 - 6 = 20.4 - 6
\]
\[
4a = 14.4
\]
Step 3: Solve for the Variable
Divide both sides by 4 to isolate a:
\[
a = \frac{14.4}{4} = 3.6
\]
Final answer:
\[
\boxed{a = 3.6}
\]
Tips for Using The Distributive Property Effectively
Applying the distributive property correctly is crucial for solving algebraic equations. Here are some tips:
1. Always Distribute to All Terms
When you see parentheses multiplied by a coefficient, distribute that coefficient to every term inside the parentheses to avoid mistakes.
2. Simplify Before and After Distribution
Simplify fractions or coefficients before distributing when possible. After distributing, combine like terms to streamline solving.
3. Check Your Work
After solving for the variable, substitute your answer back into the original equation to verify correctness.
Additional Examples of Using The Distributive Property
Practicing with similar equations can solidify your understanding.
Example 1: Solve 3/4 (8x - 4) = 9
Solution:
Distribute 3/4:
\[
\frac{3}{4} \times 8x = \frac{3 \times 8x}{4} = \frac{24x}{4} = 6x
\]
\[
\frac{3}{4} \times (-4) = -3
\]
Rewrite:
\[
6x - 3 = 9
\]
Add 3 to both sides:
\[
6x = 12
\]
Divide by 6:
\[
x = 2
\]
Answer: x = 2
---
Example 2: Solve 5/2 (2y + 6) = 15
Solution:
Distribute 5/2:
\[
\frac{5}{2} \times 2y = \frac{5 \times 2y}{2} = 5y
\]
\[
\frac{5}{2} \times 6 = \frac{5 \times 6}{2} = 15
\]
Rewrite:
\[
5y + 15 = 15
\]
Subtract 15 from both sides:
\[
5y = 0
\]
Divide both sides by 5:
\[
y = 0
\]
---
Conclusion: Mastering The Distributive Property
Using the distributive property to solve equations like 2/3 (6a + 9) = 20.4 is an essential skill in algebra. It simplifies complex expressions, helps isolate variables, and paves the way for straightforward solutions. Remember to distribute carefully, simplify your expressions, and verify your answers to ensure accuracy. With practice, applying the distributive property will become second nature, empowering you to solve a wide range of algebraic equations confidently.
By mastering this property, you lay a solid foundation for more advanced algebra topics and problem-solving techniques. Keep practicing with different equations, and soon you'll find solving algebraic expressions to be a smooth and manageable process.
