Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Ready to dive into the world of equations? Today, we're going to tackle the equation: $\frac{d}{5}-\frac{d-6}{10}=\frac{7}{10}$. Don't worry if it looks a bit intimidating at first – we'll break it down into easy-to-follow steps. Solving equations is a fundamental skill in mathematics, and once you get the hang of it, you'll be solving all sorts of problems with confidence. The process involves isolating the variable (in this case, 'd') on one side of the equation. We'll use a combination of algebraic manipulations, such as multiplying, adding, subtracting, and dividing, to achieve this. The goal is to get 'd' by itself, which will reveal the solution to the equation. Throughout this guide, we'll explain each step in detail, ensuring you grasp the underlying principles. We'll also cover how to check your answer to make sure it's correct – a crucial step for building confidence in your problem-solving abilities. Let's get started and unravel this equation together! Remember, practice makes perfect, so don't be afraid to try similar problems on your own. You've got this!

Step 1: Clearing the Fractions

Okay guys, the first thing we want to do when we see fractions in an equation is to get rid of them. It's often easier to work with whole numbers, right? So, let's look at our equation again: $\frac{d}{5}-\frac{d-6}{10}=\frac{7}{10}$. To eliminate the fractions, we need to find the least common denominator (LCD) of all the denominators. In this case, the denominators are 5 and 10. The LCD of 5 and 10 is 10. Now, we'll multiply every term in the equation by 10. This is super important: we have to multiply every term to keep the equation balanced. So we do the following:

• $10 \times \frac{d}{5} - 10 \times \frac{d-6}{10} = 10 \times \frac{7}{10}$

Let's simplify this step by step. Multiplying the first term gives us:

• $10 \times \frac{d}{5} = \frac{10d}{5} = 2d$

Next, the second term:

• $10 \times \frac{d-6}{10} = \frac{10(d-6)}{10} = d - 6$

And finally, the third term:

• $10 \times \frac{7}{10} = \frac{70}{10} = 7$

So, our equation now becomes: $2d - (d - 6) = 7$. See how much cleaner that looks? Remember, the key here is to multiply every term by the LCD. This ensures that the fractions are cleared without changing the equation's fundamental meaning. Always double-check your work to ensure you've multiplied each term correctly. This is a crucial step in simplifying and solving the equation, setting the stage for the rest of the process. If you're struggling, try writing out each step slowly and deliberately, focusing on the details. The goal is to eliminate fractions and make the equation easier to manipulate.

Why Clearing Fractions Matters

Clearing the fractions simplifies the equation and makes it easier to solve. Working with whole numbers often reduces the chances of making arithmetic errors. Additionally, it streamlines the subsequent steps of isolating the variable. Imagine trying to solve the equation without getting rid of the fractions – you'd constantly have to deal with fractions in your calculations, which can quickly become cumbersome and increase the likelihood of mistakes. Clearing fractions is a strategic move that sets you up for success. It is a fundamental algebraic technique that simplifies the entire solution process, making it more efficient and reducing the potential for calculation errors. It transforms the equation into a more manageable form, which is easier to manipulate and solve. By eliminating the fractions, you are paving the way for a smoother, more straightforward solution, allowing you to focus on the core algebraic operations needed to isolate the variable and arrive at the correct answer.

Step 2: Simplifying the Equation

Alright, now that we've cleared the fractions, let's simplify our equation further. We have $2d - (d - 6) = 7$. The first thing to do is to get rid of those parentheses. Remember that minus sign in front of the parentheses? That means we need to distribute the -1 to both terms inside the parentheses. So, $-(d - 6)$ becomes $-d + 6$. Our equation now looks like this: $2d - d + 6 = 7$. Next, we're going to combine like terms. On the left side of the equation, we have $2d$ and $-d$. Combining these, we get $d$ (because $2d - d = d$). So, the equation becomes: $d + 6 = 7$. See how things are becoming more manageable? Simplifying the equation by removing parentheses and combining like terms is all about making it easier to isolate our variable, 'd'. It reduces the complexity and helps us to focus on the essential steps required to solve for 'd'. Always double-check your work when simplifying. Ensure you've correctly distributed any negative signs and accurately combined the like terms. This stage is fundamental in setting up the final steps for solving the equation. Remember, each step brings us closer to isolating 'd' and finding the solution. Don't rush; take your time and make sure you understand the underlying principles.

The Importance of Order of Operations

In this step, it's very important to follow the correct order of operations (PEMDAS/BODMAS). This is a crucial part in simplifying the equation correctly. The order of operations dictates the sequence in which calculations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Because we had parentheses, we dealt with those first by distributing the negative sign. Then, we combined like terms, which simplified the equation further. It ensures that everyone arrives at the same correct answer, regardless of the complexity of the equation. Failure to adhere to the order of operations can easily lead to incorrect answers. So, always keep the order of operations in mind, paying close attention to parentheses and exponents, before tackling multiplication, division, addition, or subtraction. This ensures that you simplify the equation accurately and ultimately solve for the correct answer.

Step 3: Isolating the Variable

Okay, we're getting close to the finish line! Our simplified equation is $d + 6 = 7$. Our goal now is to isolate 'd' – that is, get 'd' all by itself on one side of the equation. To do this, we need to get rid of the '+ 6' on the left side. The way to get rid of a '+ 6' is to subtract 6 from both sides of the equation. Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced. So, we'll subtract 6 from both sides:

• $d + 6 - 6 = 7 - 6$

This simplifies to:

• $d = 1$

And there you have it, folks! We've solved for 'd'. We've found that $d = 1$. The essence of isolating a variable lies in using inverse operations to undo the operations performed on the variable. In this case, we used subtraction to remove the '+ 6'. Remember to perform the same operation on both sides of the equation, this maintains the equation's balance. Always double-check your steps to ensure you're using the correct inverse operations and performing them correctly on both sides. This step is about systematically removing any terms or coefficients that are attached to the variable, leaving you with a clear solution. By isolating the variable, we find the value that satisfies the original equation.

The Balancing Act

The fundamental principle in solving equations is maintaining the balance. Think of the equation like a seesaw. To keep the seesaw balanced, you must perform the same operation on both sides. If you add something to one side, you must add the same amount to the other side. If you subtract something from one side, you must subtract the same amount from the other side. This ensures that the equation remains true. This is the foundation of all algebraic manipulations. This balance is critical because it ensures that you don't change the equation's meaning. This principle lets us manipulate the equation while preserving its integrity. It allows us to systematically simplify and solve the equation without altering its underlying truth. The importance of maintaining balance cannot be overstated because it is essential to finding the correct solution.

Step 4: Checking the Answer

This is where the real fun begins, guys! We've solved for 'd', but how do we know if our answer, $d = 1$, is correct? That's where checking our answer comes in. This is a super important step because it gives you confidence in your solution and helps you catch any errors you might have made along the way. To check our answer, we're going to substitute the value of 'd' (which is 1) back into the original equation: $\frac{d}{5}-\frac{d-6}{10}=\frac{7}{10}$. So, we replace every 'd' with '1':

• $\frac{1}{5}-\frac{1-6}{10}=\frac{7}{10}$

Now, let's simplify this step by step. First, we have:

• $\frac{1}{5}-\frac{-5}{10}=\frac{7}{10}$

Next, simplify the second fraction:

• $\frac{1}{5} + \frac{1}{2} = \frac{7}{10}$

Now, let's make sure the fractions have a common denominator. The least common denominator of 5 and 2 is 10. Let's rewrite the fractions with a denominator of 10:

• $\frac{2}{10} + \frac{5}{10} = \frac{7}{10}$

Finally, add the fractions on the left side:

• $\frac{7}{10} = \frac{7}{10}$

Since the left side equals the right side, our answer is correct! That is a success, my friends!

Why Checking is Essential

Checking the answer is not just about confirming that you got it right. It is a critical component of problem-solving. It confirms the accuracy of your work. It helps identify any errors, such as calculation mistakes or incorrect algebraic manipulations, that might have occurred during the solution process. It reinforces your understanding of the underlying concepts. By substituting the solution back into the original equation, you can see how the different parts of the equation relate to each other. This builds confidence in your skills. It is an integral part of the problem-solving process that fosters a deeper understanding of mathematical principles. It encourages a proactive approach to learning. It promotes good mathematical habits and ensures that you can trust your problem-solving abilities. It helps to ensure that you arrive at the correct solution.

Conclusion

Congratulations! You've successfully solved the equation $\frac{d}{5}-\frac{d-6}{10}=\frac{7}{10}$ and checked your answer. You’ve now equipped yourself with the skills to solve similar equations. Remember, the key is to take it step by step, simplify, isolate the variable, and always check your answer. Keep practicing, and you'll become a master of solving equations in no time! Keep up the great work, and happy solving! Remember, math is like any other skill – the more you practice, the better you become. So, don't be discouraged by challenges; embrace them as opportunities to learn and grow. Go ahead, and keep exploring the fascinating world of equations and math. You have the skills and the knowledge; now go out there and show what you can do. Always be confident in your problem-solving abilities.