Solving 7m + 11 = -4(2m + 3) A Step By Step Guide To Equivalent Equations

Hey there, math enthusiasts! Ever find yourself staring at an equation, feeling like you're trying to decipher an ancient code? Well, today, we're going to crack the code of one such equation. Let's dive into the equation 7m + 11 = -4(2m + 3) and find out which of the given options is its true equivalent. We'll break it down step by step, making sure everyone, from math newbies to seasoned pros, can follow along. So, grab your pencils, and let's get started!

Decoding the Equation: 7m + 11 = -4(2m + 3)

Before we jump into solving, let's first understand what we're dealing with. The equation 7m + 11 = -4(2m + 3) is a linear equation. This means it involves a variable (in this case, 'm') raised to the power of 1. Our goal is to simplify this equation and isolate 'm' on one side to find its value. To do this effectively, we'll use a combination of algebraic techniques, including the distributive property, combining like terms, and performing inverse operations. Each of these steps is like a piece of the puzzle, and when we put them together, we'll reveal the solution.

Step 1: Distribute the -4

The first thing we need to do is get rid of those parentheses. Remember the distributive property? It's our best friend here! We need to multiply the -4 outside the parentheses by each term inside. So, -4 multiplied by 2m is -8m, and -4 multiplied by +3 is -12. This transforms our equation into:

7m + 11 = -8m - 12

See? We've already made progress! The equation looks a bit less intimidating now that we've removed the parentheses. It's like defusing the first part of a complex mechanism – we're on our way to the solution!

Step 2: Gather the 'm' Terms

Now, let's get all the 'm' terms on one side of the equation. A common strategy is to move the term with the smaller coefficient, but really, you can choose either. To keep things positive (and maybe a bit simpler), let's add 8m to both sides of the equation. This will eliminate the -8m on the right side and bring it over to the left. Remember, what we do to one side of the equation, we must do to the other to keep it balanced. It's like a mathematical seesaw – we need to keep both sides equal.

Adding 8m to both sides gives us:

7m + 8m + 11 = -8m + 8m - 12

Simplifying this, we get:

15m + 11 = -12

We're getting closer! The equation is becoming more streamlined, and 'm' is gradually being isolated.

Step 3: Isolate the 'm' Term

Our next mission is to isolate the term with 'm'. We need to get rid of that +11 on the left side. How do we do that? By using the inverse operation! The inverse of addition is subtraction, so we'll subtract 11 from both sides of the equation. This will cancel out the +11 on the left and move the constant term to the right side.

Subtracting 11 from both sides gives us:

15m + 11 - 11 = -12 - 11

Simplifying this, we have:

15m = -23

Bingo! Look familiar? This equation matches one of our answer choices. We've found a possible equivalent equation, but let's not stop here. We should continue solving for 'm' to see if it matches up with any other options after further simplification.

Step 4: Solve for 'm'

To finally solve for 'm', we need to undo the multiplication. The 15 is multiplying 'm', so we'll do the opposite – we'll divide both sides of the equation by 15. This will isolate 'm' on the left side, giving us its value.

Dividing both sides by 15, we get:

15m / 15 = -23 / 15

Simplifying, we find:

m = -23 / 15

So, the value of 'm' is -23/15. This is the solution to our original equation. Now, let's go back to our answer choices and see which one is equivalent to the steps we've taken.

Evaluating the Answer Choices

We've simplified the equation 7m + 11 = -4(2m + 3) and arrived at two key forms: 15m = -23 and m = -23/15. Now, let's compare these with the options provided to see which one matches.

A. -15m = -23 B. 15m = -23 C. -m = 1 D. -m = -1

Looking at our simplified equation 15m = -23, we can immediately see that option B matches perfectly! This confirms that 15m = -23 is indeed an equivalent equation to the original.

But what about the other options? Let's quickly examine them:

• Option A, -15m = -23, is close, but the sign of the 'm' term is incorrect. If we were to divide both sides by -15, we'd get m = 23/15, which is not our solution.
• Options C and D, -m = 1 and -m = -1, are further off. These equations would give us m = -1 and m = 1, respectively, neither of which matches our solution of m = -23/15.

Therefore, the only equation that is equivalent to the given equation is option B.

The Final Verdict

After carefully simplifying the equation 7m + 11 = -4(2m + 3) and comparing it with the answer choices, we've confidently arrived at the solution.

The correct answer is B. 15m = -23

We successfully navigated the steps of distribution, combining like terms, and isolating the variable to find the equivalent equation. Remember, math equations are like puzzles – each step is a piece that fits together to reveal the final picture. Keep practicing, and you'll become a master puzzle solver in no time!

Understanding the Underlying Principles

Now that we've solved the equation, let's take a moment to appreciate the fundamental principles at play. Understanding these principles will not only help you solve similar equations but also deepen your overall grasp of algebra.

The Distributive Property

The distributive property is a cornerstone of algebraic manipulation. It allows us to simplify expressions involving parentheses by multiplying a term outside the parentheses by each term inside. In our equation, we used it to transform -4(2m + 3) into -8m - 12. The distributive property is based on the idea that multiplication distributes over addition (and subtraction). It's a powerful tool for expanding and simplifying expressions.

Combining Like Terms

Combining like terms is another essential skill in algebra. Like terms are terms that have the same variable raised to the same power (or are constants). We combined 7m and 8m to get 15m. Combining like terms helps us to simplify equations by reducing the number of terms, making them easier to work with. It's like organizing your toolbox – grouping similar tools together makes it easier to find what you need.

Inverse Operations

Inverse operations are pairs of operations that undo each other. Addition and subtraction are inverse operations, as are multiplication and division. We used inverse operations to isolate 'm' – we subtracted 11 to undo the addition and divided by 15 to undo the multiplication. The concept of inverse operations is fundamental to solving equations because it allows us to "peel away" the operations surrounding the variable, one by one, until we have the variable by itself.

Maintaining Balance

Throughout the process of solving the equation, we emphasized the importance of maintaining balance. What we do to one side of the equation, we must do to the other. This is because an equation is a statement of equality – the two sides are equal. Any operation we perform must preserve this equality. It's like balancing a scale – if you add weight to one side, you must add the same weight to the other to keep it balanced.

Practice Makes Perfect

Solving equations is a skill that improves with practice. The more equations you solve, the more comfortable you'll become with the process. You'll start to recognize patterns, anticipate steps, and solve equations more efficiently. So, don't be discouraged if you find it challenging at first. Keep practicing, and you'll see your skills grow.

Here are some tips for practicing:

• Start with simpler equations: Build your confidence by solving equations with fewer steps and simpler numbers.
• Work through examples: Follow along with examples, paying attention to each step and why it's being taken.
• Try different types of equations: Practice solving equations with different structures, such as those with fractions, decimals, or variables on both sides.
• Check your answers: Always check your solutions by plugging them back into the original equation to make sure they work.
• Don't be afraid to ask for help: If you're stuck, ask a teacher, tutor, or friend for assistance.

Remember, every equation you solve is a step forward. Keep practicing, and you'll become an equation-solving expert!

Real-World Applications

You might be wondering, "When will I ever use this in the real world?" Well, the truth is, linear equations are used in countless applications, from everyday problem-solving to complex scientific models. Here are just a few examples:

Budgeting and Finance

Linear equations can help you manage your budget, calculate loan payments, and determine investment returns. For instance, you can use an equation to calculate how much money you need to save each month to reach a financial goal.

Measurement and Conversions

Many measurement conversions can be expressed as linear equations. For example, the formula to convert Celsius to Fahrenheit is a linear equation: F = (9/5)C + 32. You can use this equation to convert temperatures or to solve for the Celsius equivalent of a given Fahrenheit temperature.

Physics and Engineering

Linear equations are fundamental in physics and engineering. They are used to model motion, forces, and electrical circuits. For example, Ohm's Law (V = IR), which relates voltage, current, and resistance in an electrical circuit, is a linear equation.

Data Analysis

Linear equations can be used to model relationships between variables in data analysis. For example, you might use a linear equation to model the relationship between advertising spending and sales revenue.

These are just a few examples, but they illustrate the wide range of applications for linear equations. The skills you develop in solving equations are valuable tools that can be applied in many different contexts.

Conclusion: Embracing the Power of Equations

We've journeyed through the equation 7m + 11 = -4(2m + 3), dissected its components, and emerged victorious with the correct answer: B. 15m = -23. But more importantly, we've explored the underlying principles and techniques that make equation-solving possible. We've seen how the distributive property, combining like terms, and inverse operations work together to simplify and solve equations.

Remember, equations are not just abstract symbols; they are powerful tools for modeling and solving real-world problems. By mastering the art of equation-solving, you're equipping yourself with a valuable skill that can be applied in countless situations.

So, keep practicing, keep exploring, and keep embracing the power of equations! You've got this!