Solve For R A Step-by-Step Guide

Are you struggling with solving for 'r' in the equation 1/3 + 5/3 r - r = 2r? Don't worry, guys! This comprehensive guide will walk you through the step-by-step process, making it super easy to understand. We'll break down the equation, simplify it, and isolate 'r' to find the solution. Whether you're a student tackling algebra or just brushing up on your math skills, this article is your go-to resource. So, let's dive in and conquer this equation together!

Understanding the Basics of Linear Equations

Before we jump into solving the equation, let's quickly recap what a linear equation is. In simple terms, a linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because when you graph them, they form a straight line. The equation we're dealing with, 1/3 + 5/3 r - r = 2r, fits this description, making it a linear equation with 'r' as the variable.

The beauty of linear equations lies in their simplicity and predictability. They follow a set of rules that, once mastered, make solving them a breeze. The main goal is always the same: to isolate the variable on one side of the equation. This means getting 'r' all by itself on either the left or right side of the equals sign. To do this, we use a combination of algebraic operations, ensuring that we perform the same operation on both sides of the equation to maintain balance. This balance is crucial because any change on one side must be mirrored on the other to keep the equation true.

When solving linear equations, you'll often encounter terms with the variable (like 5/3 r or -r) and constant terms (like 1/3). The key is to group like terms together. This usually involves adding or subtracting terms to move them across the equals sign. Remember, when you move a term from one side to the other, you change its sign. For example, if you move +2r from the right side to the left side, it becomes -2r. This simple rule is a cornerstone of solving linear equations, and mastering it will significantly improve your ability to tackle more complex problems.

Another essential aspect of solving linear equations is simplifying fractions and combining coefficients. In our equation, we have fractions like 1/3 and 5/3, which might seem daunting at first. However, dealing with fractions becomes much easier when you find a common denominator. In this case, the denominator is already 3 for most terms, which simplifies things considerably. When combining terms with 'r', you'll be working with coefficients, which are the numbers multiplying the variable. For instance, in 5/3 r, the coefficient is 5/3. Combining these coefficients correctly is vital for accurately isolating 'r' and finding the solution. With these foundational concepts in mind, we're well-equipped to tackle the given equation and solve for 'r'.

Step-by-Step Solution for 1/3 + 5/3 r - r = 2r

Now, let's get down to the nitty-gritty and solve the equation 1/3 + 5/3 r - r = 2r step by step. Grab your pencil and paper, guys, and let's work through this together!

Step 1: Combine Like Terms on the Left Side

The first thing we want to do is simplify the left side of the equation. We have two terms with 'r': 5/3 r and -r. To combine them, we need to think of -r as -1r. So, we're essentially adding 5/3 r and -1r. To do this, we need a common denominator, which is already 3. We can rewrite -1r as -3/3 r. Now, we can add the coefficients: 5/3 - 3/3 = 2/3. Therefore, 5/3 r - r simplifies to 2/3 r. Our equation now looks like this: 1/3 + 2/3 r = 2r.

Step 2: Get All 'r' Terms on One Side

Next, we want to get all the terms with 'r' on one side of the equation. Let's choose the right side for this. To do this, we'll subtract 2/3 r from both sides of the equation. This gives us: 1/3 + 2/3 r - 2/3 r = 2r - 2/3 r. On the left side, 2/3 r and -2/3 r cancel each other out, leaving us with just 1/3. On the right side, we need to subtract 2/3 r from 2r. To do this, we rewrite 2r as 6/3 r (since 2 = 6/3). Now we have 6/3 r - 2/3 r, which equals 4/3 r. Our equation is now: 1/3 = 4/3 r.

Step 3: Isolate 'r'

Now comes the final step: isolating 'r'. We have 1/3 = 4/3 r. To get 'r' by itself, we need to get rid of the 4/3 that's multiplying it. We can do this by multiplying both sides of the equation by the reciprocal of 4/3, which is 3/4. So, we multiply both sides by 3/4: (3/4) * (1/3) = (3/4) * (4/3 r). On the left side, (3/4) * (1/3) simplifies to 1/4. On the right side, (3/4) * (4/3) cancels out, leaving us with just 'r'. So, we have: 1/4 = r.

Step 4: State the Solution

We've done it! We've successfully isolated 'r' and found the solution. The solution to the equation 1/3 + 5/3 r - r = 2r is r = 1/4. You can also write this as r = 0.25 if you prefer decimals.

By following these steps, you can confidently solve similar linear equations. Remember, the key is to break the problem down into manageable steps, combine like terms, and isolate the variable. With practice, you'll become a pro at solving for 'r' and other variables in no time!

Common Mistakes to Avoid When Solving for 'r'

When solving for 'r' (or any variable, really), it's easy to stumble upon some common pitfalls. Let's highlight these mistakes so you can steer clear of them and ensure you're on the right track. Avoiding these errors will not only save you time but also boost your confidence in your problem-solving abilities. So, let's dive in and see what these common mistakes are!

Mistake 1: Forgetting to Distribute Correctly

One of the most frequent errors occurs when dealing with expressions that involve distribution. This usually happens when you have a number multiplying a set of terms inside parentheses. For instance, if you had an equation like 2(r + 3) = 10, you need to correctly distribute the 2 to both 'r' and 3. That means multiplying 2 by 'r' to get 2r and multiplying 2 by 3 to get 6. The equation then becomes 2r + 6 = 10. Failing to distribute properly can lead to incorrect solutions. Always double-check that you've multiplied the term outside the parentheses by every term inside.

Mistake 2: Incorrectly Combining Like Terms

Combining like terms is a fundamental step in solving equations, but it's also a place where mistakes can creep in. Remember, you can only combine terms that have the same variable and exponent. For example, you can combine 3r and 5r because they both have 'r' to the power of 1, but you can't combine 3r and 5r^2 because the exponents are different. Also, pay close attention to the signs (positive or negative) in front of the terms. For example, if you have 3r - 5r, the result is -2r, not 2r. A simple sign error can throw off your entire solution, so be meticulous when combining like terms.

Mistake 3: Not Performing the Same Operation on Both Sides

The golden rule of solving equations is to maintain balance. Whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side. This ensures that the equation remains equal. For instance, if you subtract 2 from the left side, you must also subtract 2 from the right side. If you divide the right side by 3, you must also divide the left side by 3. Forgetting this rule can lead to a completely wrong answer. Always double-check that you've applied the same operation to both sides of the equation.

Mistake 4: Sign Errors

Sign errors are sneaky little devils that can trip up even the most careful problem solvers. These errors often occur when moving terms from one side of the equation to the other. Remember, when you move a term across the equals sign, you change its sign. For example, if you have r + 5 = 10 and you want to move the 5 to the right side, it becomes -5. So the equation becomes r = 10 - 5. Failing to change the sign correctly is a common mistake. Pay extra attention to the signs of the terms, especially when adding or subtracting them.

Mistake 5: Forgetting the Order of Operations

The order of operations (often remembered by the acronym PEMDAS/BODMAS) dictates the sequence in which you perform mathematical operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Ignoring this order can lead to incorrect simplification. For example, in the expression 2 + 3 * r, you should multiply 3 by 'r' before adding 2. If you add 2 and 3 first, you'll get the wrong result. Always follow the order of operations to ensure accurate calculations.

By being aware of these common mistakes and taking the time to double-check your work, you can significantly reduce the chances of making errors when solving for 'r'. Remember, practice makes perfect, so keep working at it, and you'll become a master equation solver in no time!

Practice Problems to Sharpen Your Skills

Now that we've covered the step-by-step solution and common mistakes, it's time to put your knowledge to the test! Solving practice problems is the best way to solidify your understanding and build confidence in your skills. So, let's dive into some exercises that will help you sharpen your abilities to solve for 'r'. Grab your pencil and paper, guys, and let's get to work!

Problem 1: Solve for r in the equation 2r + 5 = 11

This is a classic linear equation that tests your ability to isolate 'r'. Start by subtracting 5 from both sides of the equation to get 2r = 6. Then, divide both sides by 2 to find the value of 'r'. You should end up with r = 3. This problem reinforces the basic steps of solving linear equations.

Problem 2: Solve for r in the equation 3(r - 2) = 9

This problem introduces the concept of distribution. First, distribute the 3 across the terms inside the parentheses: 3 * r = 3r and 3 * -2 = -6. The equation becomes 3r - 6 = 9. Next, add 6 to both sides to get 3r = 15. Finally, divide both sides by 3 to solve for 'r'. The solution is r = 5. This problem emphasizes the importance of correctly applying the distributive property.

Problem 3: Solve for r in the equation 4r - 7 = 2r + 1

This equation requires you to combine like terms from both sides. Start by subtracting 2r from both sides to get 2r - 7 = 1. Then, add 7 to both sides to get 2r = 8. Finally, divide both sides by 2 to find 'r'. The solution is r = 4. This problem tests your ability to manipulate equations and keep the balance while moving terms around.

Problem 4: Solve for r in the equation (2/3)r + 1 = 5

This problem involves a fraction, so you'll need to handle it carefully. First, subtract 1 from both sides to get (2/3)r = 4. To isolate 'r', multiply both sides by the reciprocal of 2/3, which is 3/2. So, (3/2) * (2/3)r = (3/2) * 4. This simplifies to r = 6. This problem reinforces how to deal with fractions in equations.

Problem 5: Solve for r in the equation 5(r + 2) - 3r = 16

This problem combines distribution and combining like terms. First, distribute the 5: 5 * r = 5r and 5 * 2 = 10. The equation becomes 5r + 10 - 3r = 16. Next, combine like terms: 5r - 3r = 2r. So, the equation is now 2r + 10 = 16. Subtract 10 from both sides to get 2r = 6. Finally, divide both sides by 2 to find 'r'. The solution is r = 3. This problem challenges you to apply multiple skills in one equation.

By working through these practice problems, you'll reinforce your understanding of the steps involved in solving for 'r'. Remember to take your time, double-check your work, and learn from any mistakes you make. With consistent practice, you'll become more confident and proficient in solving linear equations. Keep up the great work, and you'll be an algebra whiz in no time!

Conclusion: Mastering the Art of Solving for 'r'

Congratulations, guys! You've made it to the end of this comprehensive guide on solving for 'r'. We've covered a lot of ground, from understanding the basics of linear equations to working through step-by-step solutions, avoiding common mistakes, and tackling practice problems. By now, you should have a solid understanding of how to approach and solve equations for 'r', no matter how they're presented.

The journey of mastering algebra, like any skill, is a marathon, not a sprint. It requires patience, persistence, and a willingness to learn from your mistakes. Don't get discouraged if you stumble along the way. Every mistake is a learning opportunity, a chance to deepen your understanding and refine your skills. The key is to keep practicing, keep asking questions, and keep pushing yourself to grow.

Remember, the techniques and strategies we've discussed in this guide are not limited to solving for 'r' alone. They are fundamental principles that apply to a wide range of algebraic problems. By mastering these basics, you're building a strong foundation for more advanced math concepts. So, the time and effort you invest in understanding linear equations will pay dividends in your future mathematical endeavors.

As you continue your math journey, remember to break down complex problems into smaller, more manageable steps. Just like we did with our equation, 1/3 + 5/3 r - r = 2r, you can tackle any challenge by breaking it down and approaching it systematically. Identify the key steps, apply the appropriate techniques, and double-check your work. This methodical approach will serve you well not just in math, but in any problem-solving situation.

Finally, always remember that math is not just about numbers and equations; it's about logical thinking, problem-solving, and critical reasoning. These are skills that are valuable in all aspects of life. So, embrace the challenge, enjoy the process, and celebrate your successes along the way. Keep practicing, keep learning, and keep growing. You've got this!