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. \$2p - 7 = 13$[/tex\]C. $3p - 7 = 13 - P$D. $4p - 7 = 13$

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:

$$3p - 7 + p = 13$$

Our goal is to simplify this equation and solve for the variable $p$.

Step 1: Combine Like Terms

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

$$3p + p = 4p$$

So, the equation becomes:

$$4p - 7 = 13$$

The Resulting Equation

After combining like terms, we have arrived at the resulting equation:

$$4p - 7 = 13$$

This is the equation we will use to solve for the variable $p$.

Comparing with the Options

Now, let's compare our resulting equation with the options provided:

A. $p - 7 = 13 - 3p$ B. $2p - 7 = 13$ C. $3p - 7 = 13 - p$ D. $4p - 7 = 13$

Our resulting equation matches option D.

Conclusion

In this article, we have solved a linear equation step by step, combining like terms and arriving at the correct solution. We have also compared our resulting equation with the options provided and found that it matches option D.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps and combining like terms, you can simplify linear equations and solve for the variable.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not combining like terms: Make sure to combine like terms to simplify the equation.
• Not following the order of operations: Follow the order of operations (PEMDAS) to ensure that you evaluate expressions correctly.
• Not checking your work: Double-check your work to ensure that you have arrived at the correct solution.

By avoiding these common mistakes, you can ensure that you solve linear equations correctly and confidently.

Practice Problems

To practice solving linear equations, try the following problems:

1. Solve the equation: $2x + 5 = 11$
2. Solve the equation: $x - 3 = 7$
3. Solve the equation: $4x + 2 = 14$

Remember to follow the steps outlined in this article and combine like terms to simplify the equation.

Conclusion

Introduction

In our previous article, we explored the process of solving linear equations step by step. We combined like terms and arrived at the correct solution. In this article, we will answer some 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(s) is 1. In other words, it is 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: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I combine like terms?

A: To combine like terms, we need to identify the terms that have the same variable(s) and the same exponent(s). We can then add or subtract these terms to simplify the expression.

For example, in the equation:

$$3x + 2x = 5x$$

we can combine the like terms $3x$ and $2x$ to get $5x$.

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. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2.

For example, the equation:

$$x^2 + 4x + 4 = 0$$

is a quadratic equation because the highest power of the variable $x$ is 2.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, we need to follow the same steps as we would for a linear equation without fractions. We can then multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

For example, in the equation:

$$\frac{1}{2}x + 3 = 5$$

we can multiply both sides of the equation by 2 to get:

$$x + 6 = 10$$

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable. A system of linear equations, on the other hand, is a set of two or more linear equations with the same variables.

For example, the system of linear equations:

$$x + y = 4$$

$$2x - 3y = 5$$

is a system of two linear equations with the variables $x$ and $y$.

Conclusion

Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and combining like terms, you can simplify linear equations and solve for the variable. Remember to avoid common mistakes and practice solving linear equations to build your confidence and skills.

Practice Problems

To practice solving linear equations, try the following problems:

1. Solve the equation: $\frac{1}{3}x + 2 = 5$
2. Solve the equation: $x - 2 = 7$
3. Solve the system of linear equations:

$$x + y = 4$$

$$2x - 3y = 5$$

Remember to follow the steps outlined in this article and combine like terms to simplify the equation.