Consider The Equation $3p - 7 + P = 13$. What Is The Resulting Equation After The First Step In The Solution?A. $p - 7 = 13 - 3p$ B. $20 - 7 = 13$ C. $3p - 7 = 13 - P$ D. $4p - 7 = 13$

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, step by step, to understand the process and arrive at the correct solution.

The Given Equation

The given equation is 3p−7+p=133p - 7 + p = 13. Our goal is to simplify this equation and solve for the variable pp.

Step 1: Combine Like Terms

The first step in solving the equation is to combine like terms. In this case, we have two terms with the variable pp, which are 3p3p and pp. We can combine these terms by adding their coefficients.

3p + p = 4p

Now, let's rewrite the original equation with the combined like terms:

4p - 7 = 13

The Resulting Equation

After combining like terms, the resulting equation is 4p−7=134p - 7 = 13. This is the equation we will work with in the next step.

Comparison with Answer Choices

Let's compare the resulting equation with the answer choices:

  • A. p−7=13−3pp - 7 = 13 - 3p
  • B. 20−7=1320 - 7 = 13
  • C. 3p−7=13−p3p - 7 = 13 - p
  • D. 4p−7=134p - 7 = 13

The resulting equation 4p−7=134p - 7 = 13 matches answer choice D.

Conclusion

In this article, we solved a linear equation step by step, starting with the given equation 3p−7+p=133p - 7 + p = 13. We combined like terms to simplify the equation and arrived at the resulting equation 4p−7=134p - 7 = 13. This equation matches answer choice D, which is the correct solution.

Tips and Tricks

When solving linear equations, remember to:

  • Combine like terms to simplify the equation
  • Isolate the variable on one side of the equation
  • Check your solution by plugging it back into the original equation

By following these steps and tips, you can become proficient in solving linear equations and tackle more complex math problems with confidence.

Common Mistakes to Avoid

When solving linear equations, be careful not to:

  • Forget to combine like terms
  • Make errors when isolating the variable
  • Neglect to check your solution

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics and engineering: Linear equations are used to model the motion of objects and solve problems involving forces and energies.
  • Economics: Linear equations are used to model the behavior of economic systems and solve problems involving supply and demand.
  • Computer science: Linear equations are used to solve problems involving algorithms and data structures.

By understanding how to solve linear equations, you can apply this knowledge to a wide range of fields and industries.

Practice Problems

Try solving the following linear equations:

  1. 2x+5=112x + 5 = 11
  2. x−3=7x - 3 = 7
  3. 4x+2=144x + 2 = 14

Use the steps outlined in this article to solve each equation and check your solutions.

Conclusion

Introduction

In our previous article, we explored the process of solving linear equations step by step. Now, we'll answer some frequently asked questions about solving linear equations to help you better understand the concept.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it's an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

  • The highest power of the variable(s) is 1.
  • The equation can be written in the form ax + b = c.
  • The equation does not contain any exponents or roots.

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to combine like terms. This involves adding or subtracting the coefficients of the same variable(s) to simplify the equation.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, follow these steps:

  1. Combine like terms to simplify the equation.
  2. Add or subtract the same value to both sides of the equation to eliminate the constant term.
  3. Divide both sides of the equation by the coefficient of the variable to solve for 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(s) is 1, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example:

  • Linear equation: 2x + 3 = 5
  • Quadratic equation: x^2 + 4x + 4 = 0

Q: Can I use algebraic methods to solve linear equations?

A: Yes, algebraic methods can be used to solve linear equations. Some common algebraic methods include:

  • Adding or subtracting the same value to both sides of the equation
  • Multiplying or dividing both sides of the equation by the same value
  • Using inverse operations to isolate the variable

Q: Are there any real-world applications of linear equations?

A: Yes, linear equations have numerous real-world applications, including:

  • Physics and engineering: Linear equations are used to model the motion of objects and solve problems involving forces and energies.
  • Economics: Linear equations are used to model the behavior of economic systems and solve problems involving supply and demand.
  • Computer science: Linear equations are used to solve problems involving algorithms and data structures.

Q: How can I practice solving linear equations?

A: To practice solving linear equations, try the following:

  • Use online resources, such as Khan Academy or Mathway, to practice solving linear equations.
  • Work through practice problems in a textbook or online resource.
  • Create your own practice problems and solve them.

Conclusion

Solving linear equations is a fundamental skill that requires practice and patience. By understanding the steps involved in solving linear equations and practicing regularly, you can become proficient in solving linear equations and tackle more complex math problems with confidence. Remember to combine like terms, isolate the variable, and check your solution to ensure accuracy and reliability. With practice and persistence, you can master the art of solving linear equations and apply this knowledge to a wide range of fields and industries.