Solving 0.4r - 0.13 - 0.7r = 3r - 5.74 Step-by-Step Solution

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Introduction

In this article, we will provide a step-by-step guide on how to solve for the variable r in the given linear equation: 0.4r - 0.13 - 0.7r = 3r - 5.74. This equation is a fundamental example of algebraic problem-solving, which involves simplifying, combining like terms, and isolating the variable to find its value. Understanding how to solve such equations is crucial for various mathematical and real-world applications. Whether you're a student learning algebra or someone looking to brush up on your math skills, this guide will walk you through each step clearly and concisely.

Understanding the Equation

Before diving into the solution, let's break down the equation 0.4r - 0.13 - 0.7r = 3r - 5.74. This equation is a linear equation because the highest power of the variable r is 1. It involves decimals, which might seem daunting at first, but we will tackle them systematically. The left side of the equation (0.4r - 0.13 - 0.7r) contains terms with r and a constant term, while the right side (3r - 5.74) also includes a term with r and a constant. Our goal is to isolate r on one side of the equation to determine its value.

Step-by-Step Solution

Step 1: Combine Like Terms on the Left Side

The first step is to simplify each side of the equation by combining like terms. On the left side, we have two terms with r: 0.4r and -0.7r. Combining these gives us:

0.4r - 0.7r = -0.3r

So, the left side of the equation simplifies to:

-0.3r - 0.13

Now, our equation looks like this:

-0.3r - 0.13 = 3r - 5.74

Step 2: Move Terms with r to One Side

Next, we want to get all the terms with r on one side of the equation. To do this, we can add 0.3r to both sides of the equation. This will eliminate the r term from the left side:

-0.3r - 0.13 + 0.3r = 3r - 5.74 + 0.3r

Simplifying both sides, we get:

-0.13 = 3.3r - 5.74

Step 3: Move Constant Terms to the Other Side

Now, we want to isolate the term with r. To do this, we need to move the constant term (-5.74) from the right side to the left side. We can accomplish this by adding 5.74 to both sides:

-0.13 + 5.74 = 3.3r - 5.74 + 5.74

Simplifying both sides, we have:

5.61 = 3.3r

Step 4: Isolate r

To find the value of r, we need to isolate it. We can do this by dividing both sides of the equation by the coefficient of r, which is 3.3:

5.61 / 3.3 = 3.3r / 3.3

Performing the division, we get:

r = 1.7

Verification

To ensure that our solution is correct, we can substitute r = 1.7 back into the original equation and check if both sides are equal:

0.4(1.7) - 0.13 - 0.7(1.7) = 3(1.7) - 5.74

Let's calculate each side:

Left side:

0. 4(1.7) = 0.68

0.7(1.7) = 1.19

0. 68 - 0.13 - 1.19 = -0.64

Right side:

3(1.7) = 5.1

5.1 - 5.74 = -0.64

Since both sides of the equation are equal (-0.64 = -0.64), our solution r = 1.7 is correct.

Common Mistakes to Avoid

When solving equations like this, it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

  1. Incorrectly Combining Like Terms: Make sure you only combine terms that have the same variable or are constants. For example, you can combine 0.4r and -0.7r, but you can't combine 0.4r and -0.13.
  2. Sign Errors: Pay close attention to the signs (positive and negative) of the terms. A simple sign error can throw off your entire solution.
  3. Not Distributing Properly: If the equation involves parentheses, make sure you distribute any coefficients correctly. For example, if you have 2(r + 3), you need to multiply both r and 3 by 2.
  4. Dividing or Multiplying Incorrectly: When isolating the variable, ensure you perform the same operation on both sides of the equation. Also, be careful with division, especially when dealing with decimals.
  5. Forgetting to Verify: Always substitute your solution back into the original equation to verify that it's correct. This simple step can help you catch errors and ensure your answer is accurate.

By being mindful of these common mistakes, you can increase your accuracy and confidence in solving algebraic equations.

Real-World Applications

Solving linear equations like 0.4r - 0.13 - 0.7r = 3r - 5.74 isn't just an abstract mathematical exercise; it has numerous real-world applications. Here are a few examples:

  1. Financial Planning: Imagine you are managing a budget and need to calculate how much money to allocate to different expenses. Linear equations can help you determine how much you can spend on various categories while staying within your budget.
  2. Physics and Engineering: Many physical phenomena can be modeled using linear equations. For example, calculating the distance traveled by an object at a constant speed involves solving a linear equation. Engineers use these equations to design structures, circuits, and systems.
  3. Chemistry: Balancing chemical equations often requires solving linear equations. Stoichiometry, the study of the quantitative relationships between reactants and products in chemical reactions, relies heavily on this skill.
  4. Business and Economics: Linear equations are used in supply and demand analysis, cost-benefit analysis, and various economic models. They help businesses make informed decisions about pricing, production, and investment.
  5. Everyday Problem Solving: From calculating grocery bills to determining the best deals while shopping, linear equations are a fundamental tool for solving everyday problems. They help you make informed decisions and manage resources effectively.

Understanding and being able to solve linear equations is a valuable skill that extends far beyond the classroom. It empowers you to analyze situations, make predictions, and solve problems in a wide range of contexts.

Conclusion

In summary, we have successfully solved the equation 0.4r - 0.13 - 0.7r = 3r - 5.74 by following a step-by-step approach. We combined like terms, moved variables to one side, isolated r, and found that r = 1.7. We also verified our solution by substituting it back into the original equation. This process highlights the importance of algebraic manipulation and careful attention to detail. Remember to practice solving similar equations to reinforce your understanding and skills. Mastering these techniques will not only help you in mathematics but also in numerous real-world applications where problem-solving and analytical thinking are essential.

By understanding the steps and principles involved in solving linear equations, you can approach more complex problems with confidence. Keep practicing, and you'll find that these skills become second nature. Whether you're tackling academic challenges or real-world scenarios, the ability to solve equations is a valuable asset.