Solving 5x - 4 = 4 - 5x A Step-by-Step Guide

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Given the equation 5x - 4 = 4 - 5x, what is the simplified result?

In the realm of mathematics, solving equations is a fundamental skill. One common type of equation is a linear equation, which involves variables raised to the power of one. In this comprehensive guide, we will delve into the step-by-step process of solving the linear equation 5x - 4 = 4 - 5x. By mastering these techniques, you'll gain a solid foundation for tackling more complex mathematical problems.

To effectively solve the equation 5x - 4 = 4 - 5x, we will follow a series of logical steps, ensuring that we isolate the variable 'x' on one side of the equation. This will lead us to the solution, which represents the value of 'x' that satisfies the equation. Our goal is to simplify the equation and manipulate it algebraically to arrive at the final answer.

Understanding the Basics of Linear Equations

Before we dive into the specifics of solving 5x - 4 = 4 - 5x, let's first establish a clear understanding of linear equations. A linear equation is an algebraic equation in which the highest power of the variable is one. These equations can be represented in the general form of ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. The key characteristic of a linear equation is that it represents a straight line when graphed on a coordinate plane.

Linear equations are prevalent in various fields of mathematics and science. They are used to model real-world scenarios, such as calculating distances, determining costs, and predicting outcomes. Proficiency in solving linear equations is therefore essential for students and professionals alike.

Step-by-Step Solution

Now, let's embark on the journey of solving the equation 5x - 4 = 4 - 5x. We'll break down the process into clear, concise steps, making it easy to follow and understand.

Step 1: Combine like terms

The initial step in solving the equation involves combining like terms. Like terms are those that have the same variable raised to the same power. In our equation, we have two terms with 'x' (5x and -5x) and two constant terms (-4 and 4). Our goal is to group these terms together.

To combine like terms, we will add 5x to both sides of the equation. This eliminates the -5x term on the right side and brings all 'x' terms to the left side. The equation now becomes:

5x + 5x - 4 = 4 - 5x + 5x

Simplifying this, we get:

10x - 4 = 4

Step 2: Isolate the variable term

Next, we need to isolate the variable term (10x) on the left side of the equation. To do this, we will add 4 to both sides. This eliminates the constant term (-4) on the left side and moves it to the right side. The equation now looks like this:

10x - 4 + 4 = 4 + 4

Simplifying, we get:

10x = 8

Step 3: Solve for the variable

Now, we have a simplified equation with the variable term isolated. To solve for 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 10. This will give us the value of 'x'.

Dividing both sides by 10, we get:

10x / 10 = 8 / 10

Simplifying, we find:

x = 4/5

Solution

Therefore, the solution to the equation 5x - 4 = 4 - 5x is x = 4/5. This means that when we substitute 4/5 for 'x' in the original equation, both sides will be equal. We can verify this by plugging the value back into the original equation:

5(4/5) - 4 = 4 - 5(4/5)

Simplifying, we get:

4 - 4 = 4 - 4

0 = 0

Since both sides are equal, our solution is correct.

Alternative Methods for Solving Linear Equations

While the step-by-step method we've outlined is a reliable approach, there are alternative methods for solving linear equations. One such method is the balancing method.

The Balancing Method

The balancing method involves performing the same operations on both sides of the equation to maintain equality. The goal is to manipulate the equation until the variable is isolated on one side. This method is particularly useful for visualizing the concept of equation solving as a balancing act.

Let's apply the balancing method to our equation, 5x - 4 = 4 - 5x.

  1. Add 5x to both sides: 5x - 4 + 5x = 4 - 5x + 5x, which simplifies to 10x - 4 = 4.
  2. Add 4 to both sides: 10x - 4 + 4 = 4 + 4, which simplifies to 10x = 8.
  3. Divide both sides by 10: 10x / 10 = 8 / 10, which simplifies to x = 4/5.

As you can see, the balancing method leads us to the same solution as the step-by-step method. The choice of method often depends on individual preference and the specific equation at hand.

Common Mistakes to Avoid

Solving equations can sometimes be tricky, and it's essential to be aware of common mistakes that students often make. By avoiding these pitfalls, you can improve your accuracy and efficiency in solving equations.

  1. Incorrectly Combining Like Terms: Ensure that you only combine terms that have the same variable and exponent. For instance, you cannot combine 5x and 4 because they are not like terms.
  2. Forgetting to Distribute: When an equation involves parentheses, remember to distribute any coefficients or negative signs to all terms inside the parentheses. For example, in the expression 2(x + 3), you need to distribute the 2 to both 'x' and 3.
  3. Dividing by Zero: Remember that division by zero is undefined. If you encounter an equation where you need to divide by zero, it indicates that there is no solution or an infinite number of solutions.
  4. Sign Errors: Pay close attention to the signs (+ or -) of the terms in the equation. A simple sign error can lead to an incorrect solution.
  5. Not Checking the Solution: After solving an equation, it's always a good practice to substitute the solution back into the original equation to verify that it is correct. This helps to catch any errors made during the solving process.

By being mindful of these common mistakes, you can enhance your equation-solving skills and achieve accurate results.

Applications of Linear Equations in Real Life

Linear equations aren't just abstract mathematical concepts; they have numerous applications in our daily lives. Understanding these applications can make learning linear equations more engaging and relevant.

  1. Calculating Costs: Linear equations can be used to calculate the total cost of a product or service. For example, if a taxi charges a fixed fee of $5 plus $2 per mile, you can use a linear equation to determine the total fare for a given distance.
  2. Determining Distances: Linear equations are used in physics and engineering to calculate distances traveled. For instance, if a car travels at a constant speed, you can use a linear equation to find the distance it covers in a specific time.
  3. Predicting Outcomes: Linear equations can be used to predict future outcomes based on current trends. For example, if you know the rate at which a population is growing, you can use a linear equation to estimate the population size in the future.
  4. Financial Planning: Linear equations are used in personal finance to plan budgets and investments. For instance, you can use a linear equation to calculate the monthly payments required to pay off a loan.
  5. Recipe Scaling: When adjusting recipes for different serving sizes, linear equations can help maintain the correct proportions of ingredients. For example, if a recipe calls for 2 cups of flour for 4 servings, you can use a linear equation to determine the amount of flour needed for 8 servings.

These are just a few examples of how linear equations are used in real-world scenarios. By recognizing these applications, you can appreciate the practical value of learning to solve linear equations.

Tips for Mastering Equation Solving

Solving equations is a skill that improves with practice. Here are some tips to help you master equation solving:

  1. Practice Regularly: The more you practice solving equations, the more comfortable and confident you will become. Set aside some time each day or week to work on equation-solving problems.
  2. Start with Simple Equations: Begin with basic linear equations and gradually progress to more complex ones. This will help you build a strong foundation and avoid feeling overwhelmed.
  3. Show Your Work: Always write down each step you take when solving an equation. This makes it easier to track your progress and identify any errors you may have made.
  4. Check Your Solutions: After solving an equation, substitute the solution back into the original equation to verify that it is correct. This will help you catch mistakes and ensure accuracy.
  5. Seek Help When Needed: Don't hesitate to ask for help from teachers, classmates, or online resources if you are struggling with equation solving. There are many resources available to support your learning.

By following these tips and dedicating yourself to practice, you can develop strong equation-solving skills that will benefit you in mathematics and beyond.

In conclusion, solving the equation 5x - 4 = 4 - 5x involves a systematic approach of combining like terms, isolating the variable term, and solving for the variable. By mastering these steps, you can confidently tackle a wide range of linear equations. Remember to avoid common mistakes, explore alternative methods, and practice regularly to enhance your equation-solving skills. The solution to the equation 5x - 4 = 4 - 5x is x = 4/5, a testament to the power of algebraic manipulation in finding solutions to mathematical problems. Embrace the challenge of equation solving, and you'll unlock a valuable tool for problem-solving in various aspects of life.