1/2(x+24)=21 (distributive Property) Thanks
Understanding the equation 1/2(x+24)=21 and how it relates to the distributive property is essential for mastering algebra. This article provides a comprehensive guide to solving such equations, emphasizing the application of the distributive property, simplifying expressions, and ensuring clarity in your algebraic process. Whether you're a student seeking help or a teacher looking for resource material, this detailed explanation aims to enhance your grasp of algebraic techniques centered around the distributive property.
Understanding the Equation 1/2(x+24)=21
Before diving into solving the equation, it's important to understand its structure and what it represents. The equation involves a coefficient (1/2) multiplying a binomial expression (x + 24) and equaling 21.
Breaking Down the Components
- Coefficient: 1/2, which is a fractional multiplier.
- Expression inside parentheses: x+24, a binomial expression involving the variable x and a constant.
- Equals 21: the value that the entire expression equates to.
Significance of the Distributive Property
The distributive property allows us to multiply the coefficient across the terms inside the parentheses:
\[ a(b + c) = ab + ac \]
In this context:
\[ \frac{1}{2}(x + 24) = \frac{1}{2} \times x + \frac{1}{2} \times 24 \]
Applying the distributive property simplifies the expression and makes solving the equation more straightforward.
Applying the Distributive Property to the Equation
Let's examine the process step-by-step to understand how to utilize the distributive property effectively.
Step 1: Distribute 1/2 across the parentheses
Applying the distributive property:
\[ \frac{1}{2}(x + 24) = \frac{1}{2} \times x + \frac{1}{2} \times 24 \]
Calculations:
- \(\frac{1}{2} \times x = \frac{x}{2}\)
- \(\frac{1}{2} \times 24 = 12\)
So, the expression becomes:
\[ \frac{x}{2} + 12 \]
The original equation now is:
\[ \frac{x}{2} + 12 = 21 \]
Step 2: Isolate the variable term
Subtract 12 from both sides:
\[ \frac{x}{2} + 12 - 12 = 21 - 12 \]
Simplifies to:
\[ \frac{x}{2} = 9 \]
Step 3: Solve for x
To isolate x, multiply both sides by 2, the denominator:
\[ 2 \times \frac{x}{2} = 9 \times 2 \]
Simplifies to:
\[ x = 18 \]
Verifying the Solution
Verification ensures that the found solution is correct.
Substitute x=18 into the original equation:
Original equation:
\[ \frac{1}{2}(x + 24) = 21 \]
Substitute:
\[ \frac{1}{2}(18 + 24) = 21 \]
Calculate:
\[ \frac{1}{2} \times 42 = 21 \]
\[ 21 = 21 \]
Since both sides are equal, the solution x = 18 is verified.
Key Concepts in Solving Equations with the Distributive Property
Understanding the underlying concepts helps in solving similar algebraic equations efficiently.
1. Distributive Property
- Used to eliminate parentheses by distributing the multiplier across terms inside parentheses.
- Formula: \( a(b + c) = ab + ac \)
2. Simplification
- Combining like terms and reducing expressions to simplest form.
- Critical for isolating variables.
3. Inverse Operations
- Addition and subtraction are inverse operations, as are multiplication and division.
- Used to isolate variables after distributing and simplifying.
4. Checking Solutions
- Substituting the solution back into the original equation verifies correctness.
Common Mistakes to Avoid
When working with equations involving the distributive property, be mindful of these common errors:
- Forgetting to distribute to all terms: Ensure the multiplier applies to every term inside parentheses.
- Incorrect arithmetic with fractions: Pay attention to multiplying fractions and whole numbers accurately.
- Neglecting to simplify completely: Always reduce expressions to simplest form before solving further.
- Skipping verification: Always check your solution by substituting back into the original equation.
Extensions and Practice Problems
Practicing similar equations enhances understanding and proficiency.
Practice Problem 1:
Solve for x:
\[ \frac{3}{4}(x - 8) = 6 \]
Hint: Distribute \(\frac{3}{4}\) across \(x - 8\), then isolate x.
Practice Problem 2:
Solve:
\[ 2(x + 5) = 18 \]
Hint: Use the distributive property to expand, then solve.
Practice Problem 3:
Solve for x:
\[ \frac{1}{3}(2x + 3) = 4 \]
Hint: Distribute, then isolate x.
Summary
Mastering equations like 1/2(x+24)=21 involves understanding and applying the distributive property effectively. Key steps include distributing the coefficient across the parentheses, simplifying expressions, and using inverse operations to isolate the variable. Always verify your solutions to ensure accuracy. With consistent practice, you'll develop confidence in handling similar algebraic equations and deepen your understanding of fundamental mathematical properties.
Additional Resources
- Algebra textbooks and workbooks
- Online algebra tutorials and videos
- Practice worksheets with solutions
- Math tutoring and study groups
