Solve 3(5m-1)-7m=-9: A Step-by-Step Guide

Welcome, math enthusiasts! Today, we're diving deep into the world of algebraic equations to solve the equation $3(5 m-1)-7 m=-9$. This might look a little daunting at first glance with its parentheses and multiple terms involving 'm', but don't worry! We'll break it down piece by piece, making sure you understand each step. By the end of this guide, you'll not only know how to solve this specific equation but also gain a clearer understanding of the fundamental principles used in solving linear equations. So, grab your favorite notebook and pen, and let's get started on this mathematical adventure! We'll be using a clear, conversational tone to make learning enjoyable and effective.

Understanding the Equation: The First Step to Solving

Before we can solve the equation $3(5 m-1)-7 m=-9$, it's crucial to understand what we're dealing with. This is a linear equation in one variable, 'm'. Our goal is to find the value of 'm' that makes this equation true. Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. The equation has parentheses, which means we need to use the distributive property first. The distributive property states that $a(b+c) = ab + ac$. In our case, 'a' is 3, 'b' is 5m, and 'c' is -1. After distributing, we'll have a simpler equation with fewer terms. We'll then combine like terms – terms that have the same variable raised to the same power, or constant terms. This simplifies the equation further, bringing us closer to isolating 'm'. Remember, every step we take is designed to get 'm' all by itself on one side of the equals sign. Don't be afraid to go back and review the basic properties of algebra if you feel unsure; a strong foundation is key to mastering more complex problems. This initial understanding sets the stage for all the operations that follow, ensuring we approach the problem systematically and accurately. We're not just blindly following steps; we're understanding why each step is necessary.

Step 1: Distribute to Simplify

Our first major step to solve the equation $3(5 m-1)-7 m=-9$ is to eliminate the parentheses. We do this using the distributive property. We will multiply the 3 by each term inside the parentheses: $5m$ and $-1$. So, $3 * (5m)$ becomes $15m$, and $3 * (-1)$ becomes $-3$. The equation now transforms into: $15m - 3 - 7m = -9$. It's essential to be careful with signs here. If the number outside the parentheses were negative, we'd be multiplying by a negative, which would change the signs of the terms inside. Always double-check your multiplication, especially with negatives. This step effectively removes the grouping, making the equation more linear and easier to manipulate. Think of it as simplifying a complex landscape into a more manageable terrain. After this distribution, you'll notice that the 'm' terms are no longer grouped within parentheses, which is exactly what we want. Take a moment to look at the new form of the equation: $15m - 3 - 7m = -9$. Does it look simpler? It should! This is a testament to the power of the distributive property. We've successfully tackled the first hurdle, and the path ahead is becoming clearer.

Step 2: Combine Like Terms on One Side

Now that we've distributed, the next logical step to solve the equation $3(5 m-1)-7 m=-9$ is to combine any 'like terms' that appear on the same side of the equation. Looking at our current equation, $15m - 3 - 7m = -9$, we can see two terms that contain 'm': $15m$ and $-7m$. These are like terms because they both have the variable 'm' raised to the power of 1. We can combine them by adding or subtracting their coefficients (the numbers in front of the variable). In this case, we have $15m - 7m$, which equals $8m$. We also have a constant term, $-3$, on the left side. The right side, $-9$, is also a constant. For now, we only focus on combining terms on the left side. So, $15m - 7m$ becomes $8m$. Our equation now simplifies to: $8m - 3 = -9$. Combining like terms is a critical simplification step that reduces the number of terms we need to work with, making the equation much tidier. It's like organizing your desk before tackling a big project – getting rid of clutter makes everything more efficient. Always ensure you're combining terms that are truly 'like' – you can't combine an 'm' term with a constant term directly. This step gets us closer to isolating the 'm' term, which is our ultimate goal.

Step 3: Isolate the Variable Term

We're getting closer to finding the value of 'm' as we solve the equation $3(5 m-1)-7 m=-9$. Our current equation is $8m - 3 = -9$. The goal is to get the term with 'm' ($8m$) by itself on one side of the equation. Currently, we have $-3$ on the same side as $8m$. To move this $-3$ to the other side, we perform the inverse operation. Since it's $-3$, the inverse operation is adding $3$. We must do this to both sides of the equation to maintain balance. So, we add $3$ to the left side: $8m - 3 + 3$. This simplifies to $8m$. We also add $3$ to the right side: $-9 + 3$. This equals $-6$. Our equation now becomes: $8m = -6$. This step is all about undoing what's happening to the 'm' term. We added 3 to isolate the $8m$ term. This is a fundamental principle in algebra: whatever operation is applied to a term, you use the opposite (inverse) operation to remove it. Think of it as peeling away layers to get to the core. We've successfully isolated the term containing our variable, which is a huge accomplishment in solving any equation.

Step 4: Solve for the Variable

The final step to solve the equation $3(5 m-1)-7 m=-9$ is to get 'm' completely by itself. Our equation is now $8m = -6$. This means 8 multiplied by 'm' equals $-6$. To isolate 'm', we need to undo the multiplication by 8. The inverse operation of multiplication is division. So, we will divide both sides of the equation by 8. On the left side, we have $8m / 8$, which simplifies to just 'm'. On the right side, we have $-6 / 8$. This fraction can be simplified. Both $-6$ and $8$ are divisible by 2. So, $-6 / 8$ simplifies to $-3 / 4$. Therefore, $m = -3/4$. Congratulations! You've successfully solved the equation. This final step is where we reveal the value of our unknown. Always simplify your fractions when possible, as it presents the answer in its most concise form.

Verification: Ensuring Your Solution is Correct

It's always a good practice to verify your solution to any equation. This means plugging the value you found for 'm' back into the original equation to see if it holds true. Our original equation was $3(5 m-1)-7 m=-9$, and we found that $m = -3/4$. Let's substitute $-3/4$ for 'm' in the original equation:

$3(5 * (-3/4) - 1) - 7 * (-3/4) = -9$

First, calculate the term inside the parentheses: $5 * (-3/4) = -15/4$.

Now, the expression inside the parentheses is: $-15/4 - 1$. To subtract 1, we need a common denominator, so $1 = 4/4$. This becomes $-15/4 - 4/4 = -19/4$.

So, the equation is now: $3 * (-19/4) - 7 * (-3/4) = -9$.

Next, perform the multiplications:

$3 * (-19/4) = -57/4$

$-7 * (-3/4) = 21/4$

Now, substitute these values back into the equation:

$-57/4 + 21/4 = -9$

Combine the fractions on the left side (they already have a common denominator):

$(-57 + 21) / 4 = -36 / 4$

Finally, simplify the fraction:

$-36 / 4 = -9$

So, we have $-9 = -9$. This is a true statement! Our solution $m = -3/4$ is correct. Verification is a powerful tool that builds confidence in your answers and helps catch any mistakes you might have made along the way. It's a small extra step that can save you a lot of trouble.

Conclusion: Mastering Algebraic Equations

We've successfully navigated the journey to solve the equation $3(5 m-1)-7 m=-9$, breaking it down into manageable steps: distribution, combining like terms, isolating the variable term, and finally, solving for the variable. Remember, these principles apply to a vast array of algebraic problems. The key is to remain systematic, careful with your arithmetic (especially signs!), and to always remember the golden rule of equations: whatever you do to one side, you must do to the other. Practice is your best friend when it comes to mastering algebra. The more equations you solve, the more intuitive these steps will become. Don't hesitate to tackle different types of linear equations to build your confidence and skill set. If you ever feel stuck, go back to the basics – understanding the properties of equality and operations is fundamental.

For further exploration into solving equations and other mathematical concepts, you can check out resources like Khan Academy, which offers comprehensive lessons and practice exercises on a wide range of math topics. Another excellent resource is Math is Fun, which explains mathematical concepts in an easy-to-understand and engaging way. Happy solving!