Which Two Equations Have The Same Solution As The Equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$? Select From The Following Options: $2.3p - 10.1 = 6.4p - 4$ $2.3p - 10.1 = 6.49p - 4$ $230p - 1010 = 650p - 400 - P$ $23p - 101 = 65p - 40 - P$ $2.3p - 14.1 = 6.4p - 4$

In the realm of mathematics, particularly when dealing with algebra, the concept of equivalent equations is fundamental. Equivalent equations are equations that, despite potentially looking different, share the same solution set. This means that if a certain value for a variable satisfies one equation, it will also satisfy its equivalent forms. Identifying equivalent equations is crucial for simplifying problems and finding solutions more efficiently. This article delves into how we can use properties of equality to rewrite equations while preserving their solutions. We'll focus on the given equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$ and explore which of the provided options are equivalent to it. We'll achieve this by applying algebraic manipulations such as combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by the same non-zero value. These operations, rooted in the properties of equality, ensure that the balance of the equation is maintained, and thus, the solution set remains unchanged.

H2: Properties of Equality: The Foundation of Equation Manipulation

Before diving into the specific equation, let's briefly recap the properties of equality. These properties are the bedrock upon which we perform algebraic manipulations to rewrite equations without altering their solutions.

• Addition Property of Equality: If $a = b$, then $a + c = b + c$ for any real number $c$. This means we can add the same value to both sides of an equation without changing its solution.
• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$ for any real number $c$. Similarly, subtracting the same value from both sides maintains the equality.
• Multiplication Property of Equality: If $a = b$, then $ac = bc$ for any real number $c$. Multiplying both sides by the same non-zero value preserves the solution.
• Division Property of Equality: If $a = b$, and $c ≠ 0$, then $a/c = b/c$. Dividing both sides by the same non-zero value also maintains the equality.
• Distributive Property: $a(b + c) = ab + ac$. This property allows us to expand expressions by multiplying a term across a sum or difference.
• Combining Like Terms: Terms with the same variable raised to the same power can be combined. For example, $2x + 3x = 5x$.

By judiciously applying these properties, we can transform equations into simpler, more manageable forms while ensuring that the solutions remain consistent. In the context of our problem, we'll use these properties to rewrite the given equation and compare it with the provided options.

H2: Analyzing the Original Equation: $2.3p - 10.1 = 6.5p - 4 - 0.01p$

Our starting point is the equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$. The first step in simplifying this equation is to combine like terms. On the right-hand side, we have two terms involving 'p': $6.5p$ and $-0.01p$. Combining these, we get:

$6.5p - 0.01p = (6.5 - 0.01)p = 6.49p$

Substituting this back into the original equation, we obtain:

$2.3p - 10.1 = 6.49p - 4$

This simplified form is crucial because it allows us to directly compare the original equation with the given options. We've used the basic principle of combining like terms, which is an application of the distributive property and addition/subtraction properties, to arrive at a more concise representation of the original equation. This highlights the importance of carefully examining each equation and identifying opportunities for simplification before attempting more complex manipulations. The ability to recognize and combine like terms is a foundational skill in algebra, and it often serves as the first step in solving equations efficiently. In the subsequent sections, we will compare this simplified form with the provided options to determine which equations share the same solution set as the original.

H2: Evaluating the Options: Identifying Equivalent Equations

Now, let's evaluate the given options to determine which equations have the same solution as $2.3p - 10.1 = 6.5p - 4 - 0.01p$, which we simplified to $2.3p - 10.1 = 6.49p - 4$.

H3: Option 1: $2.3p - 10.1 = 6.4p - 4$

This equation is very similar to our simplified equation. The only difference is the coefficient of $p$ on the right-hand side: $6.4p$ instead of $6.49p$. Since these coefficients are different, the two equations are not equivalent. They will have different solutions for $p$.

H3: Option 2: $2.3p - 10.1 = 6.49p - 4$

This equation is identical to the simplified form of our original equation. Therefore, this equation is equivalent to the original and will have the same solution set.

H3: Option 3: $230p - 1010 = 650p - 400 - p$

To determine if this equation is equivalent, we can try to manipulate the original equation to match this form. Let's start with the original equation: $2.3p - 10.1 = 6.5p - 4 - 0.01p$. We already simplified this to $2.3p - 10.1 = 6.49p - 4$.

Now, let's multiply both sides of the simplified equation by 100 to eliminate the decimals:

$100(2.3p - 10.1) = 100(6.49p - 4)$

This gives us:

$230p - 1010 = 649p - 400$

Notice that this is very close to Option 3, but the right side has $649p$ instead of $650p - p$. However, $650p - p$ is indeed equal to $649p$. So, Option 3 can be rewritten as:

$230p - 1010 = 649p - 400$

Which matches our manipulated equation. Therefore, Option 3 is equivalent to the original equation.

H3: Option 4: $23p - 101 = 65p - 40 - p$

Similar to Option 3, let's manipulate our simplified equation ($2.3p - 10.1 = 6.49p - 4$) to see if it matches this form. This time, instead of multiplying by 100, let's multiply both sides of the original equation ($2.3p - 10.1 = 6.5p - 4 - 0.01p$) by 10:

$10(2.3p - 10.1) = 10(6.5p - 4 - 0.01p)$

This gives us:

$23p - 101 = 65p - 40 - 0.1p$

This equation is close to Option 4, but it has $-0.1p$ instead of $-p$. Therefore, Option 4 is not equivalent to the original equation.

H3: Option 5: $2.3p - 14.1 = 6.4p - 4$

This equation differs from our original simplified equation in the constant term on the left-hand side. It has $-14.1$ instead of $-10.1$. Therefore, this equation is not equivalent to the original.

H2: Conclusion: Identifying the Equivalent Equations

Through careful analysis and application of the properties of equality, we've determined that two of the provided options have the same solution as the original equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$. These equivalent equations are:

• Option 2: $2.3p - 10.1 = 6.49p - 4$
• Option 3: $230p - 1010 = 650p - 400 - p$

Understanding how to rewrite equations while preserving their solutions is a critical skill in algebra. By mastering the properties of equality and practicing algebraic manipulation, you can confidently tackle a wide range of equation-solving problems. Remember to always look for opportunities to simplify equations by combining like terms and applying the distributive property before attempting more complex transformations. This systematic approach will help you avoid errors and arrive at the correct solution more efficiently.

In summary, identifying equivalent equations involves a combination of algebraic manipulation and careful comparison. By understanding the underlying principles and practicing diligently, you can develop the skills necessary to excel in this fundamental area of mathematics.

H2: Final Answer

The two equations that have the same solution as $2.3p - 10.1 = 6.5p - 4 - 0.01p$ are:

• $2.3p - 10.1 = 6.49p - 4$
• $230p - 1010 = 650p - 400 - p$