Use properties to rewrite the given equation. Which equations have the same solution as $2.3 p-10.1=6.5 p-4-0.01 p$ ? Select two options.

$2.3 p-10.1=6.4 p-4$

$2.3 p-10.1=6.49 p-4$

$230 p-1010=650 p-400-p$

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

$2.3 p-14.1=6.4 p-4$

The two equations that have the same solution as the given equation are Option 2: 2.3 p − 10.1 = 6.49 p − 4 and Option 3: 230 p − 1010 = 650 p − 400 − p .
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Answered by Anonymous | 2025-07-04

Simplify the original equation: 2.3 p − 10.1 = 6.5 p − 4 − 0.01 p becomes 2.3 p − 10.1 = 6.49 p − 4 .
Option 2 is identical to the simplified equation: 2.3 p − 10.1 = 6.49 p − 4 .
Simplify Option 3: 230 p − 1010 = 650 p − 400 − p becomes 230 p − 1010 = 649 p − 400 .
The two equations with the same solution are 2.3 p − 10.1 = 6.49 p − 4 and 230 p − 1010 = 650 p − 400 − p .

Explanation

Understanding the Problem We are given the equation 2.3 p − 10.1 = 6.5 p − 4 − 0.01 p and asked to find two equivalent equations from the given options.

Simplifying the Equation First, simplify the right side of the original equation by combining like terms: 2.3 p − 10.1 = 6.5 p − 4 − 0.01 p 2.3 p − 10.1 = ( 6.5 − 0.01 ) p − 4 2.3 p − 10.1 = 6.49 p − 4

Identifying Equivalent Equations Now, let's examine the given options:


Option 1: 2.3 p − 10.1 = 6.4 p − 4 Option 2: 2.3 p − 10.1 = 6.49 p − 4 Option 3: 230 p − 1010 = 650 p − 400 − p Option 4: 23 p − 101 = 65 p − 40 − p Option 5: 2.3 p − 14.1 = 6.4 p − 4
Comparing the simplified equation 2.3 p − 10.1 = 6.49 p − 4 with the options, we see that Option 2 is identical to the simplified equation. Therefore, Option 2 is one of the correct answers.

Verifying Option 3 Next, let's simplify Option 3: 230 p − 1010 = 650 p − 400 − p 230 p − 1010 = 649 p − 400

Now, let's multiply both sides of the simplified original equation 2.3 p − 10.1 = 6.49 p − 4 by 100: 100 ( 2.3 p − 10.1 ) = 100 ( 6.49 p − 4 ) 230 p − 1010 = 649 p − 400
Since this is the same as the simplified Option 3, Option 3 is also a correct answer.

Final Answer Therefore, the two equations that have the same solution as the given equation are: 2.3 p − 10.1 = 6.49 p − 4 230 p − 1010 = 650 p − 400 − p

Examples
When solving equations in physics or engineering, it's often necessary to manipulate equations to isolate variables or simplify expressions. This problem demonstrates how to rewrite an equation while preserving its solution, a crucial skill in these fields. For instance, if you're modeling the motion of an object and need to solve for its velocity, you might start with a complex equation involving multiple terms. By applying algebraic properties, you can rewrite the equation into a simpler form that makes it easier to solve for the velocity. This process ensures that the solution you obtain is accurate and consistent with the original equation.

Answered by GinnyAnswer | 2025-07-04