Equivalent Equation To 5(2p-6)=40 Step-by-Step Solution
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of identifying the equation equivalent to $5(2p-6)=40$, offering a step-by-step guide to simplify and manipulate algebraic expressions. We'll explore the core principles of equation solving, emphasizing the importance of maintaining balance and applying the distributive property effectively. Through clear explanations and detailed examples, we aim to equip you with the knowledge and confidence to tackle similar mathematical challenges. Let's embark on this journey to unravel the complexities of algebraic equations and discover the equivalent form of the given expression.
In mathematics, equivalent equations are equations that have the same solutions. This means that if a value of a variable satisfies one equation, it will also satisfy the other equivalent equation. The concept of equivalence is crucial in algebra because it allows us to transform complex equations into simpler forms that are easier to solve. To create equivalent equations, we apply operations that maintain the balance of the equation. These operations include adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number. Another key technique is the use of the distributive property, which helps us to eliminate parentheses and simplify expressions.
Why are equivalent equations important? They provide a pathway to simplify problems, making them solvable. Imagine trying to directly solve a complicated equation versus solving a much simpler, equivalent form. The latter is almost always easier. This simplification is not just about ease; it also reduces the chances of making errors during the solving process. Equivalent equations are foundational in algebra, used in everything from basic equation solving to more advanced topics like calculus. They ensure that we can manipulate expressions without changing the solution, which is a powerful tool in any mathematical context.
To determine which equation is equivalent to $5(2p-6)=40$, we must systematically simplify the given equation. This process involves applying the distributive property and then performing basic arithmetic operations to both sides of the equation to maintain its balance. Let's break down the solution step-by-step:
Step 1: Apply the Distributive Property The given equation is $5(2p-6)=40$. The first step is to eliminate the parentheses by applying the distributive property, which states that $a(b+c) = ab + ac$. Applying this to our equation, we multiply 5 by both terms inside the parentheses:
5 * (2p) - 5 * (6) = 40$ which simplifies to $10p - 30 = 40$. **Step 2: Compare the Simplified Equation with the Options** Now that we have simplified the original equation to $10p - 30 = 40$, we can compare this result with the options provided: * A. $7p - 1 = 40
- B. $10p - 6 = 40$
- C. $7p - 11 = 40$
- D. $10p - 30 = 40$
By comparing our simplified equation $10p - 30 = 40$ with the options, we can clearly see that option D, $10p - 30 = 40$, matches our result.
Conclusion: Therefore, the equation $10p - 30 = 40$ is equivalent to the original equation $5(2p-6)=40$. This step-by-step simplification demonstrates how to use the distributive property to transform and solve algebraic equations.
The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It states that for any numbers a, b, and c, the following equation holds true:
This property is not just a mathematical rule; it’s a tool that simplifies complex expressions, making them easier to understand and solve. The distributive property is essential in simplifying algebraic expressions and equations. It enables us to multiply a single term by multiple terms within parentheses, effectively removing the parentheses and allowing us to combine like terms. This process is crucial for solving equations, as it transforms the equation into a more manageable form.
Understanding the Distributive Property:
To truly grasp the distributive property, let’s break down what it means. The expression a( b + c ) indicates that the term a is multiplied by the entire quantity inside the parentheses, which is the sum of b and c. The distributive property allows us to “distribute” the multiplication of a across both b and c individually. This means we multiply a by b and a by c, and then add the results together. Mathematically, this is expressed as ab + ac.
How to Apply the Distributive Property:
Applying the distributive property involves a few straightforward steps:
- Identify the term outside the parentheses (the multiplier) and the terms inside the parentheses (the addends).
- Multiply the term outside the parentheses by each term inside the parentheses.
- Write the results of the multiplications as a sum.
Common Mistakes to Avoid:
When using the distributive property, it's important to avoid common pitfalls that can lead to errors:
- Forgetting to Distribute to All Terms: Ensure that the term outside the parentheses is multiplied by every term inside the parentheses. A frequent mistake is only multiplying by the first term and neglecting the others.
- Incorrectly Handling Signs: Pay close attention to the signs of the terms. A negative term outside the parentheses will change the signs of all terms inside the parentheses when distributed.
Option D, $10p - 30 = 40$, is the correct equivalent equation because it results from correctly applying the distributive property to the original equation $5(2p - 6) = 40$. Let's revisit the step-by-step process to reinforce this understanding:
- Original Equation: We start with the equation $5(2p - 6) = 40$.
- Applying the Distributive Property: We distribute the 5 across the terms inside the parentheses. This means we multiply 5 by both 2p and -6:
-
- Simplified Equation: Combining these results, we get the simplified equation $10p - 30 = 40$.
- Comparison with Options: When we compare this simplified equation with the provided options:
- A. $7p - 1 = 40$
- B. $10p - 6 = 40$
- C. $7p - 11 = 40$
- D. $10p - 30 = 40$
It becomes clear that option D is the only equation that matches our simplified equation. The other options involve different coefficients for p or incorrect constant terms, indicating that they are not equivalent to the original equation.
Illustrating Why Other Options are Incorrect:
To further clarify why option D is correct, let's briefly examine why the other options are incorrect:
- Option A: 7p - 1 = 40 This equation does not result from correctly distributing and simplifying the original equation. The coefficient of p and the constant term are both incorrect.
- Option B: 10p - 6 = 40 This equation correctly multiplies 5 by 2p to get 10p, but it fails to correctly multiply 5 by -6. The correct result of 5 * (-6) is -30, not -6.
- Option C: 7p - 11 = 40 Like option A, this equation has both an incorrect coefficient for p and an incorrect constant term. It does not follow from any valid simplification of the original equation.
Having identified the equivalent equation, $10p - 30 = 40$, we can now solve for the variable p. Solving an equation involves isolating the variable on one side of the equation, which means performing operations to both sides of the equation until p is by itself. Let's go through the steps:
Step 1: Isolate the Term with p The first step is to isolate the term containing p, which is 10p. To do this, we need to eliminate the constant term, -30, from the left side of the equation. We can accomplish this by adding 30 to both sides of the equation. This maintains the balance of the equation and moves us closer to isolating p.
This simplifies to:
Step 2: Solve for p Now that we have isolated the term 10p, we need to solve for p. This means getting p by itself. Since p is being multiplied by 10, we can undo this multiplication by dividing both sides of the equation by 10:
This simplifies to:
Step 3: Verify the Solution To ensure that our solution is correct, we can substitute p = 7 back into the original equation $5(2p - 6) = 40$ and check if it holds true:
Since the equation holds true, our solution p = 7 is correct.
In this comprehensive guide, we've explored the process of identifying and solving equivalent equations, focusing on the equation $5(2p-6)=40$. We began by understanding the basics of equivalent equations and the importance of maintaining balance while manipulating algebraic expressions. We then walked through a step-by-step solution to find the equivalent equation, emphasizing the application of the distributive property. We delved into a detailed explanation of the distributive property, highlighting its significance in simplifying algebraic expressions and avoiding common mistakes.
We established that option D, $10p - 30 = 40$, is the correct equivalent equation and provided a clear explanation of why the other options are incorrect. Furthermore, we demonstrated how to solve the equivalent equation for p, arriving at the solution p = 7. By verifying this solution, we confirmed its accuracy and reinforced the importance of checking our work in mathematics.
Through this exploration, we've not only solved a specific problem but also gained a deeper understanding of the fundamental principles of algebra. The ability to manipulate and solve equations is a crucial skill in mathematics and many other fields. By mastering these concepts, you'll be well-equipped to tackle a wide range of mathematical challenges.
