Solving Equations: A Step-by-Step Guide For M/(m+9) = 63/22

Hey guys! Today, we're diving into the world of algebra to tackle a specific equation: m/(m+9) = 63/22. If you've ever felt a little intimidated by fractions and variables, don't worry! We're going to break this down step by step, making it super easy to understand. This isn't just about getting the right answer; it's about grasping the process so you can confidently solve similar problems in the future. So, grab your pencils, and let's get started on this mathematical adventure!

Understanding the Equation

Before we jump into solving, let’s make sure we all understand what the equation m/(m+9) = 63/22 is telling us. At its heart, this is a proportion. We have two fractions that are set equal to each other. The left side has our variable, m, both in the numerator and the denominator, while the right side consists of constants. Our goal is to isolate m – to get it by itself on one side of the equation so we can determine its value. This involves a few key algebraic techniques, which we'll explore in detail. Think of it like a puzzle; we need to carefully manipulate the pieces (terms) until we reveal the solution. What's cool about algebra is that there are rules we can follow to ensure we're always making valid moves. These rules are based on the fundamental properties of equality, which allow us to perform the same operation on both sides of the equation without changing its balance. So, with a little strategy and a dash of algebraic know-how, we'll crack this equation in no time!

Why is this important?

Understanding how to solve equations like m/(m+9) = 63/22 isn't just about acing your math homework (though that's definitely a plus!). It's about building a foundation for more advanced mathematical concepts and developing critical thinking skills that are useful in all sorts of situations. When you can confidently manipulate equations, you can model real-world problems, make predictions, and solve for unknowns. For example, you might use similar techniques to calculate proportions in recipes, determine scaling factors in construction, or even understand financial formulas. Moreover, the logical reasoning and problem-solving skills you hone by working through algebraic equations translate to other areas of life. You'll become better at breaking down complex problems into smaller, more manageable steps, identifying patterns, and thinking strategically. So, while it might seem like we're just playing with numbers and variables here, we're actually building a powerful skillset that will serve you well in countless ways.

Step 1: Cross-Multiplication

The first step in solving this equation involves a technique called cross-multiplication. This is a handy trick for dealing with proportions – equations where two fractions are equal to each other. Essentially, we're going to multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. This eliminates the fractions and turns our equation into a more manageable form. So, for our equation m/(m+9) = 63/22, cross-multiplication means we'll multiply m by 22 and 63 by (m+9). This gives us a new equation: 22m = 63(m + 9). See how the fractions are gone? We've transformed our problem into a linear equation, which is something we can solve using basic algebraic principles. Cross-multiplication works because it's based on the fundamental property of equality: if a/b = c/d, then ad = bc. We're simply multiplying both sides of the equation by the denominators to clear them out. It's a neat shortcut that makes solving proportions much easier!

Why Cross-Multiplication Works

To really understand why cross-multiplication works, let's take a quick detour into the underlying math. Remember, our equation is m/(m+9) = 63/22. What we're really doing when we cross-multiply is multiplying both sides of the equation by the denominators, (m+9) and 22. Think of it like this:

• Multiply both sides by (m+9): [m/(m+9)] * (m+9) = (63/22) * (m+9) This simplifies to: m = (63/22) * (m+9)
• Now, multiply both sides by 22: m * 22 = [(63/22) * (m+9)] * 22 This simplifies to: 22m = 63(m+9)

See? We've arrived at the same result as if we had simply cross-multiplied! This shows that cross-multiplication isn't just a magic trick; it's a shortcut for a valid algebraic manipulation. By multiplying both sides of the equation by the denominators, we're maintaining the equality while clearing out the fractions. This gives us a cleaner equation to work with and brings us one step closer to solving for m.

Step 2: Distribute

Now that we've cross-multiplied and have the equation 22m = 63(m + 9), our next step is to distribute the 63 on the right side. This means we'll multiply 63 by both terms inside the parentheses: m and 9. The distributive property is a fundamental concept in algebra that allows us to simplify expressions like this. It basically says that a(b + c) = ab + ac. In our case, 63(m + 9) becomes 63 * m + 63 * 9. So, let's do the math: 63 * m is simply 63m, and 63 * 9 is 567. This gives us a new equation: 22m = 63m + 567. We've successfully expanded the right side of the equation, getting rid of the parentheses and setting us up for the next step in isolating m. Remember, the key to solving equations is to carefully apply the rules of algebra, step by step, until we have the variable by itself.

Mastering the Distributive Property

The distributive property is a cornerstone of algebra, so it's worth taking a moment to really understand it. It's not just a rule to memorize; it's a concept that shows up in all sorts of mathematical situations. The basic idea is that multiplication distributes over addition (and subtraction). This means that when you have a number multiplied by a sum (or difference) inside parentheses, you can multiply the number by each term inside the parentheses separately, and then add (or subtract) the results. Think of it like this: you're "distributing" the multiplication to each term inside the parentheses. For example, let's say we have 4(x + 3). Using the distributive property, we multiply 4 by x and 4 by 3, which gives us 4x + 12. The same principle applies with subtraction: 5(y - 2) becomes 5y - 10. The distributive property also works with more complex expressions, like those involving variables and exponents. For instance, 2x(x^2 + 4x - 1) becomes 2x^3 + 8x^2 - 2x. Mastering the distributive property is essential for simplifying expressions, solving equations, and working with polynomials. It's a fundamental tool in your algebraic arsenal!

Step 3: Isolate the Variable Terms

Our equation now looks like this: 22m = 63m + 567. The next crucial step is to isolate the variable terms, meaning we want to get all the terms containing m on one side of the equation. To do this, we need to move the 63m term from the right side to the left side. We can achieve this by subtracting 63m from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. So, subtracting 63m from both sides gives us: 22m - 63m = 63m + 567 - 63m. On the left side, 22m - 63m simplifies to -41m. On the right side, the 63m terms cancel each other out, leaving us with 567. Our equation is now simplified to -41m = 567. We're getting closer to our solution! By isolating the variable terms, we've created a situation where we have only one term containing m, making it easier to solve for its value.

The Importance of Maintaining Balance

The concept of maintaining balance in an equation is absolutely fundamental to algebra. It's the guiding principle behind every step we take when solving for a variable. Think of an equation like a perfectly balanced scale. The left side represents one weight, and the right side represents another. Our goal is to figure out the value of a specific weight (the variable) without disrupting the balance. Whenever we perform an operation on one side of the equation, we must perform the exact same operation on the other side to keep the scale balanced. This is why we can add, subtract, multiply, or divide both sides of an equation by the same number without changing its solution. If we were to perform an operation on only one side, we'd be throwing the scale out of balance, and the equation would no longer be true. This principle of balance is what allows us to manipulate equations and isolate variables. It's a powerful tool that enables us to solve all sorts of mathematical problems.

Step 4: Solve for 'm'

We've made it to the final step! Our equation is now -41m = 567. To solve for 'm', we need to isolate it completely. Currently, m is being multiplied by -41. To undo this multiplication, we'll divide both sides of the equation by -41. This is another application of the principle of balance: whatever we do to one side, we must do to the other. So, dividing both sides by -41 gives us: (-41m) / -41 = 567 / -41. On the left side, -41 divided by -41 cancels out, leaving us with just m. On the right side, 567 divided by -41 is approximately -13.83 (we'll round to two decimal places for simplicity). Therefore, our solution is m ≈ -13.83. We've successfully solved for m! This means we've found the value that, when substituted back into the original equation, makes the equation true. Pat yourself on the back – you've navigated the algebraic steps and arrived at the answer!

Verifying the Solution

It's always a good practice to verify your solution, especially in algebra. This helps ensure that you haven't made any mistakes along the way. To verify our solution, m ≈ -13.83, we'll substitute this value back into the original equation: m/(m+9) = 63/22. So, we have -13.83 / (-13.83 + 9). Let's simplify the denominator: -13.83 + 9 = -4.83. Now we have -13.83 / -4.83. If you divide -13.83 by -4.83 you get approximately 2.86. Now let's look at the other side of the original equation 63/22, if you perform the division you get approximately 2.86. Since both sides of the equation are approximately equal (remember, we rounded our value for m), our solution is verified. This gives us confidence that we've solved the equation correctly. Verification is a valuable tool for catching errors and solidifying your understanding of the problem-solving process.

Conclusion

Alright, guys! We've successfully solved the equation m/(m+9) = 63/22! We walked through each step, from cross-multiplication to isolating the variable, and finally arrived at the solution: m ≈ -13.83. Remember, the key to solving algebraic equations is to break them down into manageable steps, apply the rules of algebra consistently, and maintain balance throughout the process. Don't be afraid to tackle challenging problems; with practice and a solid understanding of the fundamentals, you can conquer any equation that comes your way. Keep practicing, keep exploring, and keep building your mathematical skills. You've got this!