Solving For J: A Step-by-Step Guide To Isolate The Variable

Hey guys! Today, we're diving into a fun little math problem where we need to solve for j. Don't worry, it's not as scary as it sounds. We'll break it down step by step so that you can easily follow along and master these types of equations. Our mission, should we choose to accept it, is to find the value of j in the equation: $-2+\frac{3}{4} j+\frac{1}{2} j=2 j$. So grab your pencils, and let's get started!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what we're dealing with. The equation $-2+\frac{3}{4} j+\frac{1}{2} j=2 j$ looks a bit complicated with the fractions, but the basic idea is that we want to isolate j on one side of the equation. This means we want to get j all by itself, so we can see what it equals. To do this, we'll use some algebraic techniques to move terms around and simplify the equation. Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced. This is a fundamental principle in algebra, ensuring that the equation remains true throughout our solving process. Think of it like a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. In our case, we're not dealing with physical weights, but with mathematical operations.

The key to effectively isolating j lies in several algebraic properties. The Addition and Subtraction Properties of Equality state that if you add or subtract the same value from both sides of an equation, the equality remains true. Similarly, the Multiplication and Division Properties of Equality dictate that multiplying or dividing both sides of an equation by the same non-zero value preserves the equality. We'll be using these properties strategically to rearrange the terms in our equation. For example, we might add a term to both sides to eliminate it from one side or multiply both sides by a constant to clear fractions. Each step we take is designed to bring us closer to having j isolated, revealing its true value. So, with these principles in mind, let's roll up our sleeves and start simplifying our equation. We'll tackle the fractions first, then group like terms, and finally, isolate j. Are you ready? Let's do this!

Step 1: Combine Like Terms

Okay, first things first, let's combine the j terms on the left side of the equation. We've got $\frac{3}{4} j$ and $\frac{1}{2} j$. To add these together, we need a common denominator. Lucky for us, we can easily turn $\frac{1}{2}$ into $\frac{2}{4}$ by multiplying both the numerator and the denominator by 2. So, our equation now looks like this: $-2+\frac{3}{4} j+\frac{2}{4} j=2 j$. Now, we can add the fractions: $\frac{3}{4} j + \frac{2}{4} j = \frac{5}{4} j$. So, the equation simplifies to: $-2+\frac{5}{4} j=2 j$.

Combining like terms is a crucial step in simplifying algebraic equations. It allows us to consolidate similar terms, making the equation more manageable and easier to solve. In our case, combining the j terms not only reduces the number of terms we're dealing with but also prepares us for the next steps in isolating j. Without this step, we'd be working with more terms, which can increase the chances of making a mistake. Think of it as decluttering your workspace before starting a project; it makes the task at hand less daunting and more organized. The process of combining like terms involves identifying terms that share the same variable (in this case, j) and then adding or subtracting their coefficients. The coefficient is the number that multiplies the variable. For instance, in the term $\frac{5}{4} j$, the coefficient is $\frac{5}{4}$. To combine like terms effectively, pay close attention to the signs (positive or negative) in front of each term. These signs dictate whether you add or subtract the coefficients. Also, remember that only terms with the same variable and exponent can be combined. For example, you can combine $\frac{5}{4} j$ and $\frac{2}{4} j$ because they both have j raised to the power of 1, but you cannot combine $\frac{5}{4} j$ with a term like $j^2$ or a constant term like -2. Mastering the art of combining like terms is a fundamental skill in algebra, and it will serve you well in solving a wide range of equations. So, let's keep this skill in mind as we move forward in solving for j. With our equation simplified, we're now ready to tackle the next step: moving all the j terms to one side.

Step 2: Move j Terms to One Side

Alright, let's get all the j terms on one side of the equation. To do this, we can subtract $\frac{5}{4} j$ from both sides. This will eliminate the j term on the left side. Our equation, $-2+\frac{5}{4} j=2 j$, now transforms into $-2+\frac{5}{4} j - \frac{5}{4} j = 2 j - \frac{5}{4} j$. Simplifying this, we get $-2 = 2 j - \frac{5}{4} j$. Now, we need to combine the j terms on the right side. To do this, let's rewrite $2j$ as $\frac{8}{4} j$ (since 2 is the same as $\frac{8}{4}$). So, we have $-2 = \frac{8}{4} j - \frac{5}{4} j$. Subtracting the fractions, we get $-2 = \frac{3}{4} j$.

Moving variable terms to one side is a critical maneuver in solving equations. The goal here is to isolate the variable we're trying to solve for, which, in this case, is j. By grouping all the j terms together, we're effectively creating a scenario where we can eventually get j by itself on one side of the equation. This process often involves performing the same operation on both sides of the equation to maintain balance, a core principle of algebraic manipulation. In our specific example, we chose to subtract $\frac{5}{4} j$ from both sides to eliminate the j term on the left. This was a strategic decision, as it simplified the equation and brought us closer to our goal. However, it's important to recognize that there might be other valid approaches. For instance, we could have chosen to subtract $2j$ from both sides instead. While this would have led to a slightly different set of intermediate steps, it would ultimately have resulted in the same solution for j. The key is to select the approach that seems most straightforward and minimizes the chances of errors. When deciding how to move terms, consider the coefficients of the variable terms. If one side has a fraction and the other has a whole number, it might be easier to move the fractional term to the side with the whole number, as we did in our example. This can help avoid dealing with mixed fractions or complex calculations later on. Also, remember that signs play a crucial role in this process. Pay close attention to whether a term is positive or negative, as this will determine whether you need to add or subtract it from both sides. By carefully managing the movement of variable terms, we set the stage for the final steps in solving for j. With the j terms now consolidated on one side, we're ready to isolate j completely and reveal its value.

Step 3: Isolate j

We're almost there! Our equation is now $-2 = \frac{3}{4} j$. To isolate j, we need to get rid of the $\frac{3}{4}$ that's multiplying it. We can do this by multiplying both sides of the equation by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$. So, we have $-2 * \frac{4}{3} = \frac{3}{4} j * \frac{4}{3}$. On the right side, $\frac{3}{4}$ and $\frac{4}{3}$ cancel each other out, leaving us with just j. On the left side, we have $-2 * \frac{4}{3} = -\frac{8}{3}$. So, our final answer is $j = -\frac{8}{3}$.

Isolating the variable is the grand finale of the solving process. It's the moment when we finally get the variable we've been chasing after all by itself, revealing its true value. In our case, we're isolating j, and the technique we employ to achieve this is the multiplication property of equality. This property allows us to multiply both sides of an equation by the same non-zero value without altering the equality. But the key here is not just multiplying by any number; we need to multiply by the reciprocal of the coefficient attached to j. The reciprocal of a fraction is simply the fraction flipped upside down. So, the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. Multiplying by the reciprocal effectively cancels out the coefficient, leaving j isolated. Think of it like having a lock and key. The coefficient is like the lock, and multiplying by its reciprocal is like using the correct key to unlock it. Once unlocked, j is free and clear, standing alone on one side of the equation. The process of multiplying both sides by the reciprocal requires careful attention to signs and fractions. Remember that multiplying a negative number by a positive number results in a negative number. Also, be sure to multiply the reciprocal by every term on both sides of the equation to maintain balance. Once you've performed the multiplication, simplify both sides to arrive at the final solution. In our example, we found that $j = -\frac{8}{3}$. This means that if we were to substitute $-\frac{8}{3}$ for j in the original equation, the equation would hold true. And that, my friends, is the ultimate test of whether we've solved for j correctly. So, with j now proudly displayed in its isolated form, we can confidently say that we've conquered this equation!

Step 4: Check Your Answer

To make sure we've nailed it, let's plug our answer, $j = -\frac{8}{3}$, back into the original equation: $-2+\frac{3}{4} j+\frac{1}{2} j=2 j$. Substituting $j = -\frac{8}{3}$, we get $-2+\frac{3}{4} * (-\frac{8}{3})+\frac{1}{2} * (-\frac{8}{3})=2 * (-\frac{8}{3})$. Let's simplify each term. $\frac{3}{4} * (-\frac{8}{3}) = -2$, $\frac{1}{2} * (-\frac{8}{3}) = -\frac{4}{3}$, and $2 * (-\frac{8}{3}) = -\frac{16}{3}$. So, our equation now looks like $-2 - 2 - \frac{4}{3} = -\frac{16}{3}$. Combining the constants on the left, we get $-4 - \frac{4}{3} = -\frac{16}{3}$. To combine these, we need a common denominator, so let's rewrite -4 as $-\frac{12}{3}$. Now we have $-\frac{12}{3} - \frac{4}{3} = -\frac{16}{3}$, which simplifies to $-\frac{16}{3} = -\frac{16}{3}$. Yay! The equation holds true, so our answer is correct!

Checking your answer is the final victory lap in the equation-solving process. It's the moment when you get to confirm that all your hard work has paid off and that you've indeed found the correct solution. Plugging your solution back into the original equation is like putting the puzzle pieces together to see if they fit. If they do, you know you've got the right picture. If they don't, it's a sign that you might need to revisit your steps and look for any potential errors. The process of checking your answer involves substituting the value you found for the variable back into the original equation. Then, you simplify both sides of the equation using the order of operations (PEMDAS/BODMAS) to see if they are equal. If the left side equals the right side, congratulations! You've solved the equation correctly. If not, don't fret. It simply means there's an error somewhere in your calculations, and it's an opportunity to learn and improve. When checking your answer, pay close attention to signs, fractions, and any other potential sources of error. It's often helpful to rewrite each step clearly and methodically to avoid making mistakes. Also, remember that checking your answer is not just about finding the right solution; it's about building confidence in your problem-solving abilities. Every time you successfully verify your answer, you reinforce your understanding of the concepts and techniques involved. So, make checking your answer a non-negotiable part of your equation-solving routine. It's the ultimate seal of approval on your mathematical prowess. With our solution now checked and confirmed, we can confidently conclude that we've solved for j! Give yourself a pat on the back, guys – you've earned it!

Conclusion

And there you have it! We've successfully solved for j in the equation $-2+\frac{3}{4} j+\frac{1}{2} j=2 j$, and we found that $j = -\frac{8}{3}$. Remember, the key to solving these types of equations is to combine like terms, move the variable terms to one side, isolate the variable, and always check your answer. Keep practicing, and you'll become a pro at solving for any variable! Keep up the great work, guys!