Solving The Linear Equation What Is The Solution?

Finding the solution to a linear equation is a fundamental concept in algebra. In this article, we will delve into the process of solving the given linear equation:

$\frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p$

We will explore the steps involved in isolating the variable p and arrive at the correct solution. Linear equations are ubiquitous in mathematics and its applications, making it crucial to understand how to solve them effectively. This article provides a comprehensive explanation and a step-by-step solution to help you master this essential skill. Let's begin by discussing the properties of linear equations and the methods we can use to find their solutions.

Understanding Linear Equations

Before diving into the specific solution, it's important to understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable's exponent is always 1, and the equation can be represented graphically as a straight line. Linear equations can involve one or more variables, but in this case, we are dealing with a single variable, p. The general form of a linear equation with one variable is:

$ax + b = c$

where x is the variable, and a, b, and c are constants. In our given equation,

$\frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p$

we can see that p is the variable, and the constants are the fractions. Solving a linear equation involves isolating the variable on one side of the equation. This is achieved by performing the same operations on both sides of the equation to maintain balance. The operations typically include addition, subtraction, multiplication, and division. The goal is to manipulate the equation until we have p equal to a constant value, which will be the solution.

Steps to Solve Linear Equations

1. Simplify Both Sides: If there are any like terms on either side of the equation, combine them. This involves adding or subtracting constants or terms with the same variable.
2. Isolate the Variable Term: Move all terms containing the variable to one side of the equation and all constants to the other side. This is usually done by adding or subtracting terms from both sides.
3. Isolate the Variable: Once you have the variable term isolated, divide both sides of the equation by the coefficient of the variable to solve for the variable.
4. Check the Solution: After finding a potential solution, substitute it back into the original equation to verify that it makes the equation true. This step is crucial to ensure that no errors were made during the solving process.

Understanding these steps is essential for solving any linear equation. Now, let's apply these steps to solve the given equation.

Step-by-Step Solution

To solve the equation

$\frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p$

we will follow the steps outlined above. Each step will be explained in detail to ensure clarity.

Step 1: Rearrange the Equation

Our goal is to isolate p on one side of the equation. To begin, we need to move the terms containing p to one side and the constants to the other. Let's subtract $\frac{3}{5}p$ from both sides of the equation:

$\frac{2}{5} + p - \frac{3}{5}p = \frac{4}{5} + \frac{3}{5}p - \frac{3}{5}p$

This simplifies to:

$\frac{2}{5} + p - \frac{3}{5}p = \frac{4}{5}$

Now, combine the p terms on the left side:

$p - \frac{3}{5}p = 1p - \frac{3}{5}p = \frac{5}{5}p - \frac{3}{5}p = \frac{2}{5}p$

So the equation becomes:

$\frac{2}{5} + \frac{2}{5}p = \frac{4}{5}$

Step 2: Isolate the Variable Term

Next, we need to isolate the term containing p. To do this, we subtract $\frac{2}{5}$ from both sides of the equation:

$\frac{2}{5} + \frac{2}{5}p - \frac{2}{5} = \frac{4}{5} - \frac{2}{5}$

This simplifies to:

$\frac{2}{5}p = \frac{4}{5} - \frac{2}{5}$

Now, subtract the fractions on the right side:

$\frac{4}{5} - \frac{2}{5} = \frac{4 - 2}{5} = \frac{2}{5}$

So the equation becomes:

$\frac{2}{5}p = \frac{2}{5}$

Step 3: Solve for p

To solve for p, we need to divide both sides of the equation by the coefficient of p, which is $\frac{2}{5}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$. So, we multiply both sides by $\frac{5}{2}$:

$\frac{5}{2} \cdot \frac{2}{5}p = \frac{5}{2} \cdot \frac{2}{5}$

This simplifies to:

$p = \frac{5 \cdot 2}{2 \cdot 5}$

$p = \frac{10}{10}$

$p = 1$

Step 4: Check the Solution

To ensure our solution is correct, we substitute p = 1 back into the original equation:

$\frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p$

$\frac{2}{5} + 1 = \frac{4}{5} + \frac{3}{5}(1)$

First, convert 1 to a fraction with a denominator of 5:

$1 = \frac{5}{5}$

So the equation becomes:

$\frac{2}{5} + \frac{5}{5} = \frac{4}{5} + \frac{3}{5}$

Now, add the fractions on both sides:

$\frac{2 + 5}{5} = \frac{4 + 3}{5}$

$\frac{7}{5} = \frac{7}{5}$

The left side equals the right side, so our solution p = 1 is correct.

Conclusion

We have successfully solved the linear equation

$\frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p$

through a step-by-step process. By rearranging the equation, isolating the variable term, and solving for p, we found that the solution is p = 1. We also verified our solution by substituting it back into the original equation and confirming that it holds true. Understanding how to solve linear equations is a critical skill in mathematics, and this detailed explanation should provide a solid foundation for tackling similar problems.

The correct answer is A. p = 1.

This exercise demonstrates the importance of following a systematic approach when solving algebraic equations. By breaking down the problem into smaller, manageable steps, we can confidently arrive at the correct solution. Linear equations form the basis for more advanced mathematical concepts, making it essential to master this fundamental skill. Whether you are a student learning algebra or someone looking to refresh your mathematical knowledge, the principles outlined here will prove invaluable. Remember to always check your solutions to ensure accuracy and build your confidence in solving equations.

Additional Tips for Solving Linear Equations

• Stay Organized: Keep your work neat and organized. Write each step clearly and align the equal signs. This will help you avoid mistakes and make it easier to review your work.
• Double-Check Your Work: After each step, double-check your calculations to ensure accuracy. A small mistake early in the process can lead to an incorrect final answer.
• Practice Regularly: The more you practice solving linear equations, the more comfortable and confident you will become. Work through a variety of problems to develop your skills.
• Understand the Properties of Equality: The properties of equality are the foundation of solving equations. Remember that you can add, subtract, multiply, or divide both sides of an equation by the same number without changing the solution.
• Use Parentheses When Necessary: If you are dealing with more complex equations involving multiple terms, use parentheses to group terms and avoid confusion.

By following these tips and practicing regularly, you can enhance your ability to solve linear equations and excel in mathematics. Linear equations are not just abstract concepts; they have practical applications in various fields, including science, engineering, economics, and computer science. Mastering linear equations opens the door to understanding and solving real-world problems.