Which Statement Is True About The Equation \[$-3(p+5)=2(p+2)+1\$\]?A. There Is No Solution. B. The Only Solution Is \[$p=\frac{2}{5}\$\]. C. The Only Solution Is \[$p=-4\$\]. D. There Are Infinitely Many Solutions.

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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 explore the equation −3(p+5)=2(p+2)+1{-3(p+5)=2(p+2)+1} and determine which statement is true about its solution. We will break down the equation step by step, using algebraic techniques to isolate the variable and find the solution.

Understanding the Equation

The given equation is a linear equation in one variable, p{p}. It is a quadratic equation in disguise, but we can simplify it by using the distributive property and combining like terms.

Step 1: Distribute the Negative 3

The first step is to distribute the negative 3 to the terms inside the parentheses:

−3(p+5)=−3p−15{-3(p+5) = -3p - 15}

Step 2: Distribute the 2

Next, we distribute the 2 to the terms inside the parentheses:

2(p+2)=2p+4{2(p+2) = 2p + 4}

Step 3: Combine Like Terms

Now, we can combine like terms on both sides of the equation:

−3p−15=2p+4+1{-3p - 15 = 2p + 4 + 1}

Simplifying the right-hand side, we get:

−3p−15=2p+5{-3p - 15 = 2p + 5}

Step 4: Isolate the Variable

To isolate the variable p{p}, we need to get all the terms with p{p} on one side of the equation. We can do this by adding 3p{3p} to both sides:

−15=5p+5{-15 = 5p + 5}

Next, we subtract 5 from both sides:

−20=5p{-20 = 5p}

Step 5: Solve for p

Finally, we can solve for p{p} by dividing both sides by 5:

p=−4{p = -4}

Conclusion

In conclusion, the solution to the equation −3(p+5)=2(p+2)+1{-3(p+5)=2(p+2)+1} is p=−4{p = -4}. This means that the only solution is p=−4{p = -4}, and there is no other value of p{p} that satisfies the equation.

Answer

The correct answer is:

C. The only solution is p=−4{p = -4}.

Discussion

This problem requires students to apply algebraic techniques to solve a linear equation. The key concepts involved are the distributive property, combining like terms, and isolating the variable. By following these steps, students can solve linear equations and understand the underlying mathematics.

Tips and Variations

  • To make this problem more challenging, students can be asked to solve a system of linear equations or a quadratic equation.
  • To make this problem easier, students can be given a simpler equation to solve, such as 2x+3=5{2x + 3 = 5}.
  • To apply this concept to real-world problems, students can be asked to solve equations that represent physical situations, such as the motion of an object or the balance of a budget.

Common Mistakes

  • Students may forget to distribute the negative 3 or the 2, leading to incorrect simplification of the equation.
  • Students may not combine like terms correctly, leading to incorrect simplification of the equation.
  • Students may not isolate the variable correctly, leading to incorrect solution of the equation.

Practice Problems

  • Solve the equation 3x−2=5x+1{3x - 2 = 5x + 1}.
  • Solve the equation 2x+3=5x−2{2x + 3 = 5x - 2}.
  • Solve the equation x−2=3x+1{x - 2 = 3x + 1}.

Introduction

In our previous article, we explored the equation −3(p+5)=2(p+2)+1{-3(p+5)=2(p+2)+1} and determined that the only solution is p=−4{p = -4}. However, we know that there are many more questions and scenarios that can arise when solving linear equations. In this article, we will answer some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (usually x) on one side of the equation. You can do this by using the following steps:

  1. Simplify the equation by combining like terms.
  2. Add or subtract the same value to both sides of the equation to get rid of any constants.
  3. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to multiply a single term to multiple terms inside parentheses. For example, 2(x+3)=2x+6{2(x + 3) = 2x + 6}.

Q: How do I use the distributive property to solve a linear equation?

A: To use the distributive property to solve a linear equation, you need to multiply the term outside the parentheses to each term inside the parentheses. For example, if you have the equation 2(x+3)=5{2(x + 3) = 5}, you can use the distributive property to rewrite it as 2x+6=5{2x + 6 = 5}.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. For example, 2x+3y=5{2x + 3y = 5} and x−2y=−3{x - 2y = -3} are a system of linear equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use the following steps:

  1. Simplify each equation by combining like terms.
  2. Use the substitution method or the elimination method to solve for one variable.
  3. Substitute the value of the variable into one of the original equations to solve for the other variable.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by substituting the value of one variable into one of the original equations.

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of linear equations by adding or subtracting the two equations to eliminate one of the variables.

Q: How do I choose between the substitution method and the elimination method?

A: To choose between the substitution method and the elimination method, you need to look at the coefficients of the variables in the two equations. If the coefficients are the same, you can use the elimination method. If the coefficients are different, you can use the substitution method.

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

A: Some common mistakes to avoid when solving linear equations include:

  • Forgetting to distribute the negative sign or the positive sign.
  • Not combining like terms correctly.
  • Not isolating the variable correctly.
  • Not checking the solution by plugging it back into the original equation.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By understanding the concepts of linear equations, the distributive property, and the substitution method and the elimination method, students can solve a wide range of linear equations and apply them to real-world situations. We hope that this Q&A guide has been helpful in answering some of the most frequently asked questions about solving linear equations.