Solve Multi-Step Equations: A Step-by-Step Guide
Hey guys! Ever stared at a math problem that looks like a tangled mess of numbers and variables, and thought, "Where do I even start?" You're not alone! Today, we're diving deep into solving multi-step equations. We'll break down how to combine like terms and use inverse operations and the properties of equality to find that elusive value of $x$. Don't worry, we'll make it super clear and easy to follow, so by the end of this, you'll be a multi-step equation whiz!
Understanding the Basics: What Are We Doing Here?
So, what exactly are we trying to achieve when we solve an equation like $-4 x-5+2 x=-11$? Our main goal is to isolate the variable, which in this case is $x$. Think of it like trying to get all the $x$ terms by themselves on one side of the equals sign and all the constant numbers on the other. To do this, we'll use a few key tools: combining like terms, inverse operations, and the properties of equality. Each of these is crucial for untangling the equation and revealing the value of $x$. We're basically playing a game of mathematical detective, piecing together clues to find the solution. It's all about reversing the operations that have been applied to $x$ in a systematic way. This process not only helps us solve the equation but also builds a strong foundation for tackling more complex algebraic challenges down the line. So, let's get our detective hats on and start unraveling this mystery!
Step 1: Combining Like Terms - Tidying Up the Equation
Alright, let's start with our equation: $-4 x-5+2 x=-11$. The first order of business is to combine like terms. What does that mean? It means grouping together terms that have the same variable (in this case, $x$) and grouping together the constant numbers. Look at the left side of our equation: $-4x - 5 + 2x$. We've got two terms with $x$: $-4x$ and $+2x$. We can combine these by simply adding their coefficients (the numbers in front of the $x$). So, $-4x + 2x$ becomes $-2x$. Now, our equation looks a bit cleaner: $-2x - 5 = -11$. See? We've already made it simpler! Combining like terms is like decluttering your workspace before you start a big project. It makes everything more manageable. Always look for terms with the same variable or terms that are just numbers. You can add or subtract them depending on their signs. This step is fundamental because it reduces the number of terms you need to deal with, making the subsequent steps of solving for $x$ much more straightforward. Remember, when combining terms, pay close attention to the signs (+ or -) in front of each term. This is where many people stumble, so take your time and be meticulous. If you have $-4x$ and $+2x$, you are essentially starting at -4 on a number line and moving 2 units to the right, which lands you at -2. So, $-4x + 2x$ correctly simplifies to $-2x$. Once you've combined all the $x$ terms and all the constant terms on each side of the equation, you'll have a simpler equation, usually in the form $ax + b = c$, which is much easier to work with.
Step 2: Using Inverse Operations - The Reversal Game
Now that we have our simplified equation, $-2x - 5 = -11$, it's time to bring in the inverse operations. Remember, every operation has an opposite, or inverse, that undoes it. Addition's inverse is subtraction, subtraction's inverse is addition, multiplication's inverse is division, and division's inverse is multiplication. Our goal is to get $x$ all by itself. Currently, $x$ is being multiplied by $-2$, and then $-5$ is being subtracted from that result. To get $x$ alone, we need to undo these operations in the reverse order they were applied (think of unwrapping a present โ you take off the outer layers first). The last operation applied was subtracting 5. So, we'll use the inverse operation: add 5 to both sides of the equation. Why both sides? That brings us to our next important concept.
Step 3: Properties of Equality - Keeping the Balance
This is where the properties of equality come into play. Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. If you add weight to one side of a scale, you have to add the same weight to the other side to maintain equilibrium. The same applies to equations! Since we decided to add 5 to the left side of our equation $-2x - 5 = -11$ to cancel out the $-5$, we must also add 5 to the right side. So, we have:
This simplifies to:
We've successfully eliminated the constant term from the side with $x$! This principle of maintaining balance is fundamental in algebra. Whether you're adding, subtracting, multiplying, or dividing, you always perform the same operation on both sides of the equals sign. This ensures that the equality remains true. Without the properties of equality, solving equations would be impossible, as the relationship between the two sides would be broken. It's the backbone of algebraic manipulation, allowing us to transform complex equations into simpler, solvable forms while guaranteeing that our solution is accurate. So, remember: balance is key!
Step 4: Finishing the Job - The Final Isolation
We're almost there, guys! Our equation is now $-2x = -6$. We've already used inverse operations and properties of equality to get this far. Now, $x$ is being multiplied by $-2$. What's the inverse operation of multiplication? You guessed it โ division! To isolate $x$, we need to divide both sides of the equation by $-2$. Applying the properties of equality again, we perform this division on both sides:
On the left side, $-2$ divided by $-2$ equals 1, leaving us with $1x$, or just $x$. On the right side, $-6$ divided by $-2$ equals $+3$. So, our final answer is:
Wait a minute! Let's recheck our math. $-4x - 5 + 2x = -11$. Combine like terms: $-2x - 5 = -11$. Add 5 to both sides: $-2x = -11 + 5 = -6$. Divide both sides by -2: $x = \frac{-6}{-2} = 3$. Phew! It seems I made a small error in my initial thought process while writing this out, which is a good reminder that even experienced folks make mistakes! Let's go back and double-check the original options given. The options were A. $x=-3$, B. $x=1$. My derived answer is $x=3$. This indicates there might be a typo in the provided options or the original equation. However, following the steps meticulously, the value of $x$ for the equation $-4 x-5+2 x=-11$ is indeed $3$. If we had to choose the closest option or if there was a typo in the problem, we'd need clarification. For the purpose of demonstrating the method, let's assume there was a typo and proceed as if one of the options should be correct. Let's re-evaluate the problem and options carefully. It seems I might have misread the options or the problem itself. Let's retrace.
Equation: $-4 x-5+2 x=-11$
Combine like terms: $-2x - 5 = -11$
Add 5 to both sides: $-2x = -11 + 5$
Divide by -2: $x = \frac{-6}{-2}$
It appears my calculations are consistent. Given the options A. $x=-3$, B. $x=1$, and the original equation, it's possible there's an error in the question's provided options. However, if we are forced to select an answer based on a potential typo, let's consider scenarios. If the equation was $-4x - 5 + 2x = 1$, then $-2x - 5 = 1$, $-2x = 6$, $x = -3$. This matches option A. If the equation was $-4x - 5 + 2x = -7$, then $-2x - 5 = -7$, $-2x = -2$, $x = 1$. This matches option B.
Let's assume for the sake of demonstrating the answer selection that the intended equation was the one that leads to option A, which is $-4x - 5 + 2x = 1$.
Option A: $x=-3$
Let's test this by plugging $x=-3$ back into the original equation: $-4(-3) - 5 + 2(-3)$. This equals $12 - 5 - 6$.
12 - 5 = 7$. $7 - 6 = 1$. So, $-4x - 5 + 2x = 1$ when $x=-3$. This does NOT equal the original right side of -11. **Option B: $x=1$** Let's test this by plugging $x=1$ back into the *original* equation: $-4(1) - 5 + 2(1)$. This equals $-4 - 5 + 2$. $-4 - 5 = -9$. $-9 + 2 = -7$. So, $-4x - 5 + 2x = -7$ when $x=1$. This also does NOT equal the original right side of -11. **Conclusion based on original equation and options:** The correct value for $x$ in the equation $-4 x-5+2 x=-11$ is $3$. None of the provided options (A. $x=-3$, B. $x=1$) match this correct solution. There appears to be an error in the question's options. However, if this were a multiple-choice test and you had to pick the *intended* answer, you would need clarification or a correction. For the sake of demonstrating the full process and providing a definite answer based on the given equation, the correct value of $x$ is **3**. Since this is not an option, we cannot definitively select A or B based on the provided equation and options. The method shown above is how you would arrive at the correct solution for the given equation. ## Final Thoughts and Practice Solving multi-step equations might seem daunting at first, but by breaking it down into these manageable steps โ combining like terms, using inverse operations, and always remembering the properties of equality โ you can tackle any problem. Practice is key, guys! The more equations you solve, the more comfortable and confident you'll become. Keep practicing, and you'll be solving complex algebraic problems in no time. Remember, math is like a muscle; the more you work it out, the stronger it gets! So, grab some more problems and get to it!