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 2.3p - 10.1 = 6.5p - 4 - 0.01p 2.3 P − 10.1 = 6.5 P − 4 − 0.01 P ? Select Two Options.A. 2.3 P − 10.1 = 6.4 P 2.3p - 10.1 = 6.4p 2.3 P − 10.1 = 6.4 P B. 2.3 P − 10.1 = 6.49 P − 4 2.3p - 10.1 = 6.49p - 4 2.3 P − 10.1 = 6.49 P − 4 C. $230p - 1010 = 650p - 400

Introduction

In mathematics, equations are a fundamental concept used to represent relationships between variables. When dealing with equations, it's essential to understand the properties that govern their manipulation. One of the most critical properties is the ability to rewrite equations in different forms without changing their solutions. In this article, we will explore how to use properties to rewrite the given equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$ and identify which equations have the same solution.

Understanding the Properties of Equations

Before we dive into rewriting the given equation, it's essential to understand the properties that govern equations. The following properties are crucial in manipulating equations:

• Addition Property: The addition property states that when we add the same value to both sides of an equation, the equation remains true.
• Subtraction Property: The subtraction property states that when we subtract the same value from both sides of an equation, the equation remains true.
• Multiplication Property: The multiplication property states that when we multiply both sides of an equation by the same non-zero value, the equation remains true.
• Division Property: The division property states that when we divide both sides of an equation by the same non-zero value, the equation remains true.

Rewriting the Given Equation

Now that we have a solid understanding of the properties that govern equations, let's rewrite the given equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$. To do this, we will use the properties mentioned above.

Step 1: Combine Like Terms

The first step in rewriting the equation is to combine like terms. In this case, we have two terms with the variable $p$ and two constant terms.

# Define the equation
equation = "2.3p - 10.1 = 6.5p - 4 - 0.01p"
combined_equation = "2.3p + (-0.01p) = 6.5p - 4"

Step 2: Simplify the Equation

Now that we have combined like terms, let's simplify the equation by adding the coefficients of the variable $p$.

# Simplify the equation
simplified_equation = "2.29p = 6.5p - 4"

Step 3: Isolate the Variable

The final step in rewriting the equation is to isolate the variable $p$. To do this, we will add $4$ to both sides of the equation and then divide both sides by the coefficient of the variable $p$.

# Isolate the variable
isolated_equation = "2.29p - 6.5p = -4"

Step 4: Simplify the Equation

Now that we have isolated the variable, let's simplify the equation by combining like terms.

# Simplify the equation
final_equation = "-4.21p = -4"

Step 5: Solve for the Variable

The final step in rewriting the equation is to solve for the variable $p$. To do this, we will divide both sides of the equation by the coefficient of the variable $p$.

# Solve for the variable
solution = "p = 0.95"

Which Equations Have the Same Solution?

Now that we have rewritten the given equation, let's identify which equations have the same solution. To do this, we will compare the final equation we obtained with the options provided.

Option A: $2.3p - 10.1 = 6.4p$

To determine if this equation has the same solution as the final equation we obtained, let's rewrite it using the same properties we used earlier.

# Rewrite the equation
rewritten_equation = "2.3p - 10.1 = 6.4p"

combined_equation = "2.3p = 6.4p + 10.1"

simplified_equation = "-4.1p = 10.1"

isolated_equation = "p = -2.46"

As we can see, the solution to this equation is not the same as the solution to the final equation we obtained.

Option B: $2.3p - 10.1 = 6.49p - 4$

To determine if this equation has the same solution as the final equation we obtained, let's rewrite it using the same properties we used earlier.

# Rewrite the equation
rewritten_equation = "2.3p - 10.1 = 6.49p - 4"

combined_equation = "2.3p = 6.49p - 4 + 10.1"

simplified_equation = "-4.19p = 6.1"

isolated_equation = "p = 1.45"

As we can see, the solution to this equation is not the same as the solution to the final equation we obtained.

Option C: $230p - 1010 = 650p - 400$

To determine if this equation has the same solution as the final equation we obtained, let's rewrite it using the same properties we used earlier.

# Rewrite the equation
rewritten_equation = "230p - 1010 = 650p - 400"

combined_equation = "230p = 650p - 400 + 1010"

simplified_equation = "-420p = 610"

isolated_equation = "p = -1.45"

As we can see, the solution to this equation is not the same as the solution to the final equation we obtained.

However, if we rewrite the equation as $2.3p - 10.1 = 6.49p - 4$, we can see that it has the same solution as the final equation we obtained.

Conclusion

In this article, we explored how to use properties to rewrite the given equation $2.3p - 10.1 = 6.5p - 4 - 0.01p$ and identify which equations have the same solution. We used the addition, subtraction, multiplication, and division properties to simplify the equation and isolate the variable. We also compared the final equation we obtained with the options provided and determined that the equation $2.3p - 10.1 = 6.49p - 4$ has the same solution as the final equation we obtained.

References

• [1] "Algebra: Structure and Method" by Richard G. Brown
• [2] "Mathematics for Dummies" by Mary Jane Sterling

Additional Resources

• [1] Khan Academy: Algebra
• [2] Mathway: Algebra Solver
  Frequently Asked Questions (FAQs) About Rewriting Equations ================================================================

Q: What are the properties of equations that govern their manipulation?

A: The properties of equations that govern their manipulation are:

• Addition Property: The addition property states that when we add the same value to both sides of an equation, the equation remains true.
• Subtraction Property: The subtraction property states that when we subtract the same value from both sides of an equation, the equation remains true.
• Multiplication Property: The multiplication property states that when we multiply both sides of an equation by the same non-zero value, the equation remains true.
• Division Property: The division property states that when we divide both sides of an equation by the same non-zero value, the equation remains true.

Q: How do I rewrite an equation using the properties of equations?

A: To rewrite an equation using the properties of equations, follow these steps:

1. Combine like terms: Combine the terms with the same variable.
2. Simplify the equation: Simplify the equation by adding or subtracting the coefficients of the variable.
3. Isolate the variable: Isolate the variable by adding or subtracting the same value from both sides of the equation.
4. Solve for the variable: Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I determine if two equations have the same solution?

A: To determine if two equations have the same solution, follow these steps:

1. Rewrite the equations: Rewrite the equations using the properties of equations.
2. Compare the equations: Compare the rewritten equations to determine if they have the same solution.

Q: What are some common mistakes to avoid when rewriting equations?

A: Some common mistakes to avoid when rewriting equations include:

• Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
• Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.
• Not solving for the variable: Failing to solve for the variable can lead to incorrect solutions.

Q: How do I use technology to help me rewrite equations?

A: There are many online tools and software programs that can help you rewrite equations, including:

• Mathway: Mathway is an online math solver that can help you rewrite equations and solve math problems.
• Khan Academy: Khan Academy is an online learning platform that offers video lessons and practice exercises on algebra and other math topics.
• Desmos: Desmos is an online graphing calculator that can help you visualize and solve math problems.

Q: What are some real-world applications of rewriting equations?

A: Some real-world applications of rewriting equations include:

• Science: Rewriting equations is used in science to model and solve real-world problems, such as the motion of objects and the behavior of chemical reactions.
• Engineering: Rewriting equations is used in engineering to design and optimize systems, such as bridges and buildings.
• Economics: Rewriting equations is used in economics to model and solve real-world problems, such as the behavior of markets and the impact of policy changes.

Conclusion

In this article, we have discussed the properties of equations that govern their manipulation, how to rewrite an equation using these properties, and some common mistakes to avoid. We have also discussed some real-world applications of rewriting equations and some online tools and software programs that can help you rewrite equations.