Simplify: Algebraic Expression With Fractions And 'j'
Hey guys! Ever get those math problems that look like a jumbled mess of fractions and variables? Today, we're going to break down one of those expressions and make it super simple. We'll be tackling this expression: (-\frac{1}{8} j+\frac{2}{3})-(\frac{5}{3} j+\frac{9}{12}). Don't worry, it's not as scary as it looks! Let’s dive right in and simplify this thing together. We'll take it one step at a time, so you can follow along easily and understand exactly what's going on.
Understanding the Expression
Okay, before we start moving things around, let's quickly understand what we're looking at. We've got two sets of terms inside parentheses. Each set contains a term with the variable 'j' and a constant term (a regular number). The key here is that we are subtracting the entire second set of parentheses from the first. This means we need to distribute the negative sign carefully. This distribution is super important, because that negative sign changes the signs of everything inside the second set of parentheses. If we miss that, the whole thing goes sideways! Remember that 'j' is just a variable, like 'x' or 'y'. It represents an unknown value. For now, we'll just treat it like any other term and combine it with its like terms. Keep in mind that the goal of simplifying any expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, -\frac{1}{8} j and \frac{5}{3} j are like terms, and \frac{2}{3} and \frac{9}{12} are like terms because they are both constants. Simplifying makes the expression easier to understand and work with, and it helps us solve equations more efficiently. So, gear up, and let’s make this expression simpler and easier to deal with.
Step 1: Distribute the Negative Sign
The first thing we need to do is get rid of those parentheses. Remember, we're subtracting the entire second group. That means we need to distribute the negative sign in front of the second set of parentheses to each term inside. So, -(\frac5}{3} j+\frac{9}{12}) becomes -\frac{5}{3} j - \frac{9}{12}. Our expression now looks like this{8} j+\frac{2}{3} - \frac{5}{3} j - \frac{9}{12}. See how the plus sign in front of \frac{5}{3}j became a minus sign, and the plus sign in front of \frac{9}{12} also became a minus sign? That's the power of the negative sign distribution! This step is crucial; messing it up will throw off the rest of the calculation. Always double-check to make sure you've correctly distributed the negative sign to every single term within the parentheses. Now that we've handled the parentheses, we are ready to start combining those like terms. We are one step closer to simplifying this expression to its core form!
Step 2: Combine the 'j' Terms
Alright, let's focus on the terms with 'j': -\frac1}{8} j and -\frac{5}{3} j. To combine these, we need a common denominator. The least common multiple of 8 and 3 is 24. So, we'll convert both fractions to have a denominator of 24. -\frac{1}{8} j becomes -\frac{3}{24} j (multiply top and bottom by 3). -\frac{5}{3} j becomes -\frac{40}{24} j (multiply top and bottom by 8). Now we can combine them{24} j - \frac{40}{24} j = -\frac{43}{24} j. So, the combined 'j' term is -\frac{43}{24} j. Remember, always find that common denominator before adding or subtracting fractions. This is one of those fundamental rules that will save you from making mistakes. We are on a roll now!
Step 3: Combine the Constant Terms
Next up, let's combine the constant terms: \frac2}{3} and -\frac{9}{12}. First, notice that \frac{9}{12} can be simplified to \frac{3}{4} by dividing both the numerator and denominator by 3. So, we have \frac{2}{3} - \frac{3}{4}. Now, we need a common denominator for 3 and 4, which is 12. \frac{2}{3} becomes \frac{8}{12} (multiply top and bottom by 4). -\frac{3}{4} becomes -\frac{9}{12} (multiply top and bottom by 3). Now we can combine them{12} - \frac{9}{12} = -\frac{1}{12}. So, the combined constant term is -\frac{1}{12}. Simplifying the fraction before finding a common denominator made this step a little easier. Always be on the lookout for opportunities to simplify fractions!
Step 4: Put It All Together
We've simplified the 'j' terms and the constant terms. Now, let's put it all together to get our final simplified expression. We found that the combined 'j' term is -\frac43}{24} j, and the combined constant term is -\frac{1}{12}. So, our simplified expression is{24} j - \frac{1}{12}. And that's it! We've successfully simplified the original expression. Remember to take it step by step, distribute negative signs carefully, and find common denominators when working with fractions. With a little practice, you'll be simplifying expressions like a pro in no time!
Final Simplified Expression
The final, simplified form of the expression is: -\frac{43}{24} j - \frac{1}{12}. This is the most concise way to represent the original expression, making it easier to understand and use in further calculations. Good job, guys, we nailed it!