Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving the equation $7(-6r - 4) = 140$. This is a classic algebra problem, and we'll break it down step by step so you can totally nail it. It might seem a bit intimidating at first, but trust me, with a little practice, you'll be solving equations like a pro. We're gonna cover all the essential steps, from distributing the 7 to isolating the variable. Let's get started!

Understanding the Problem

Solving equations is a fundamental skill in mathematics. It's all about finding the value of a variable that makes the equation true. In this case, our variable is r. The equation $7(-6r - 4) = 140$ tells us that if we multiply the expression in the parentheses by 7, we get 140. Our goal is to figure out what r must be for this to be true. Before we jump into the solution, it's crucial to understand the order of operations (PEMDAS/BODMAS). This tells us the sequence in which we need to perform the calculations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). So, we need to deal with what's inside the parentheses first, and then we can move on to the rest of the equation. The goal is to isolate the variable (r) on one side of the equation and get a numerical value on the other side. This involves a series of algebraic manipulations, using inverse operations to undo the operations applied to the variable. For example, if the variable is multiplied by a number, we'll divide both sides of the equation by that number. If a number is added to the variable, we'll subtract that number from both sides. This process maintains the equality of the equation, ensuring that the value of r we find is the correct solution. It's like a puzzle where we need to carefully rearrange the pieces (numbers and operations) to reveal the hidden answer. By mastering these steps, you'll not only solve this particular equation but also build a solid foundation for tackling more complex mathematical problems down the road. Remember, practice makes perfect! The more equations you solve, the more comfortable and confident you'll become in your abilities.

Step-by-Step Solution

Alright, let's get down to business and solve the equation $7(-6r - 4) = 140$. We'll break it down into easy-to-follow steps.

Step 1: Distribute the 7

The first step is to get rid of those parentheses. We do this by distributing the 7 to both terms inside the parentheses. That means we multiply 7 by -6r and 7 by -4. So, $7 * -6r = -42r$ and $7 * -4 = -28$. Our equation now becomes:

$-42r - 28 = 140$

It's essential to remember the rules of multiplication with negative numbers. A positive number times a negative number results in a negative number. And a positive number times a positive number is positive.

Step 2: Isolate the term with r

Next, we want to get the term with r by itself on one side of the equation. To do this, we need to get rid of the -28. Since it's being subtracted, we'll do the opposite and add 28 to both sides of the equation. This keeps the equation balanced.

$-42r - 28 + 28 = 140 + 28$

This simplifies to:

$-42r = 168$

Adding 28 to both sides cancels out the -28 on the left side and adds 28 to the right side, giving us 168. This step is crucial because it moves us closer to isolating r. Always remember to perform the same operation on both sides of the equation to maintain the equality.

Step 3: Solve for r

Now we're super close! We have $-42r = 168$. To solve for r, we need to get r by itself. Since r is being multiplied by -42, we'll do the opposite and divide both sides of the equation by -42.

$-42r / -42 = 168 / -42$

This simplifies to:

$r = -4$

And there you have it! The solution to the equation is r = -4. Dividing both sides by -42 isolates r and reveals its value. Pay close attention to the signs. A positive number divided by a negative number results in a negative number, which is essential for getting the correct answer. Now, let's move on to see if the answer matches any of the provided options.

Checking the Answer

Alright, we've got our answer (r = -4), but let's make sure it's correct. The best way to do this is to plug it back into the original equation and see if it works. This is called checking your solution. If the left side of the equation equals the right side after substituting our value for r, then we know we've got it right.

Original equation: $7(-6r - 4) = 140$

Substitute r = -4:

$7(-6 * -4 - 4) = 140$

First, let's simplify the inside of the parentheses: -6 * -4 = 24 (Remember, a negative times a negative is a positive!) So, we now have: $7(24 - 4) = 140$

Next, calculate inside the parentheses: $24 - 4 = 20$

So, we have:

$7 * 20 = 140$

And finally:

$140 = 140$

Ta-da! The left side equals the right side, which means our answer, r = -4, is correct. This step is super important because it helps us catch any silly mistakes we might have made along the way, ensuring we provide accurate solutions to the problem. Always take the time to check your answer, guys; it's a great habit to develop! If the equation doesn't balance, it means something went wrong. Go back and carefully check each step of your solution.

Matching the Answer to the Options

Okay, now that we've solved the equation and verified our answer, let's see which of the multiple-choice options matches our result. We've determined that $r = -4$. Now let's look at the options provided:

A. -1 B. 1 C. -15 D. -4

Our solution, r = -4, perfectly matches option D. Congrats, we've successfully solved the equation and found the correct answer! This step involves comparing our computed result with the offered choices, verifying that we have chosen the correct one, and boosting our confidence in our mathematical ability. When working on a multiple-choice problem, we should first derive the solution on our own before looking at the available answer options. This helps to avoid the temptation to select the wrong one accidentally. Always go through each choice, even if you are sure your answer is correct. It's always helpful to make sure you are not missing something.

Final Thoughts and Tips

Awesome job, guys! You've successfully solved the equation $7(-6r - 4) = 140$. You've learned how to distribute, isolate the variable, and check your answer. Here are a few extra tips to help you continue to excel in algebra:

• Practice, Practice, Practice: The more equations you solve, the better you'll become. Try different types of equations to broaden your understanding. Use worksheets, textbooks, or online resources to solve various problems. Consistent practice is key to mastering any mathematical concept.
• Understand the Basics: Make sure you have a solid grasp of the fundamental rules of algebra, such as the order of operations, rules of exponents, and properties of equality. Review the definitions of the basic mathematical concepts regularly to make sure you are on the right track.
• Check Your Work: Always check your answers by plugging them back into the original equation. This helps you catch any mistakes and reinforces your understanding.
• Don't Be Afraid to Ask for Help: If you get stuck, don't hesitate to ask your teacher, a tutor, or a classmate for help. There is no shame in asking for assistance; it is a critical component of the learning process. Discuss the problems and how to solve them with your teachers or classmates.
• Break It Down: When solving a complex equation, break it down into smaller, more manageable steps. This will make the process less intimidating and easier to understand.
• Stay Organized: Keep your work neat and organized. This will help you avoid making careless mistakes and will make it easier to review your work later.
• Visualize the Problem: Sometimes, drawing a diagram or using a visual representation of the equation can help you understand the problem better.

Keep up the great work, and keep practicing! You got this!