Solving For J: A Step-by-Step Guide To 2(j-12.3)-7.6=4.8

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Hey guys! Let's dive into this math problem where we need to solve for j in the equation 2(j-12.3)-7.6=4.8. Don't worry, we'll break it down step by step so it's super easy to follow. We're going to use some fundamental algebraic principles to isolate 'j' on one side of the equation. By the end of this guide, you'll not only know the answer but also understand the process. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we all understand what the equation 2(j-12.3)-7.6=4.8 is telling us. This equation is a mathematical statement that shows a relationship between numbers and a variable, which in this case is 'j'. Our main goal here is to find the value of j that makes this statement true. Think of it like a puzzle where we need to figure out what number 'j' represents.

The equation includes several parts:

  • Variable: 'j' is the unknown value we're trying to find.
  • Parentheses: (j-12.3) tells us to first subtract 12.3 from 'j'.
  • Multiplication: 2(j-12.3) means we're multiplying the result of (j-12.3) by 2.
  • Subtraction: -7.6 indicates we subtract 7.6 from the previous result.
  • Equals: =4.8 shows that the entire left side of the equation should equal 4.8.

To solve for j, we need to isolate it on one side of the equation. This means we want to get 'j' by itself, with all the other numbers on the opposite side. We do this by performing inverse operations – basically, doing the opposite of what's being done in the equation. For example, if the equation includes addition, we'll use subtraction to undo it. If it has multiplication, we'll use division, and so on.

Remember, whatever we do to one side of the equation, we must also do to the other side. This keeps the equation balanced and ensures that the value of 'j' we find is correct. So, with this understanding, we're ready to start solving. Let's move on to the first step in isolating 'j'.

Step 1: Distribute the 2

The first thing we need to do is get rid of those parentheses. We can do this by distributing the 2 across the terms inside the parentheses (j - 12.3). Distributing means we multiply the 2 by both 'j' and -12.3. This is based on the distributive property, which is a fundamental concept in algebra.

So, when we distribute the 2, we get:

2 * j = 2j

2 * -12.3 = -24.6

Now, our equation looks like this: 2j - 24.6 - 7.6 = 4.8. We've successfully removed the parentheses, which makes the equation a bit simpler to work with. The next step is to combine any like terms on the left side of the equation. Like terms are those that have the same variable raised to the same power (in this case, just the constant terms -24.6 and -7.6).

Combining like terms helps to streamline the equation further, making it easier to isolate 'j'. It's like tidying up before you start a big project – it helps to keep things organized and clear. So, let's move on to combining those constant terms and see how our equation looks then. Remember, our goal is to get 'j' all by itself on one side, and we're making good progress towards that!

Step 2: Combine Like Terms

Now that we've distributed the 2, let's combine the like terms on the left side of the equation. In our case, the like terms are the constants: -24.6 and -7.6. Combining these means we simply add them together. Remember, when you're adding two negative numbers, the result will also be negative.

So, let's add -24.6 and -7.6:

-24.6 + (-7.6) = -32.2

Now, our equation looks like this: 2j - 32.2 = 4.8. See how much simpler it's becoming? We've reduced the number of terms on the left side, bringing us closer to isolating 'j'. The next step involves getting rid of the constant term that's on the same side as 'j'. In this case, it's -32.2. To do this, we'll use the inverse operation – addition. We'll add 32.2 to both sides of the equation. This is a crucial step because it helps us move closer to having 'j' all by itself.

Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, let's move on to the next step and add 32.2 to both sides. We're on the right track to solving for j, and each step we take makes the solution clearer.

Step 3: Add 32.2 to Both Sides

To isolate the term with 'j' (which is 2j), we need to get rid of the -32.2 on the left side of the equation. We can do this by adding 32.2 to both sides. Remember, adding the same number to both sides keeps the equation balanced.

So, we add 32.2 to both sides of the equation 2j - 32.2 = 4.8:

2j - 32.2 + 32.2 = 4.8 + 32.2

On the left side, -32.2 and +32.2 cancel each other out, leaving us with just 2j. On the right side, 4.8 + 32.2 equals 37. This simplifies our equation to:

2j = 37

We're getting so close! Now, we have 2j equal to 37. This means 2 times 'j' equals 37. To find the value of a single 'j', we need to undo the multiplication. And how do we undo multiplication? That's right, we use division. So, in the next step, we'll divide both sides of the equation by 2 to finally isolate 'j'. We're almost there – just one more step to go!

Step 4: Divide Both Sides by 2

We've reached the final step in solving for j! Our equation is now 2j = 37. This tells us that 2 times 'j' equals 37. To find the value of 'j', we need to divide both sides of the equation by 2. This will undo the multiplication and leave 'j' by itself.

So, let's divide both sides by 2:

(2j) / 2 = 37 / 2

On the left side, the 2s cancel each other out, leaving us with just 'j'. On the right side, 37 divided by 2 is 18.5. So, our equation simplifies to:

j = 18.5

And there we have it! We've successfully solved for j. The value of j that makes the equation 2(j-12.3)-7.6=4.8 true is 18.5. Awesome job, guys! You've navigated through the steps of distributing, combining like terms, and using inverse operations to isolate the variable. But before we wrap up, let's quickly check our work to make sure our answer is correct.

Step 5: Check Your Solution

It's always a good idea to check your solution to make sure you haven't made any mistakes along the way. To do this, we'll substitute the value we found for 'j' (which is 18.5) back into the original equation: 2(j-12.3)-7.6=4.8. If both sides of the equation are equal after we substitute, then we know our solution is correct.

So, let's plug in 18.5 for 'j':

2(18.5 - 12.3) - 7.6 = 4.8

First, we solve the operation inside the parentheses:

  1. 5 - 12.3 = 6.2

Now, substitute this back into the equation:

2(6.2) - 7.6 = 4.8

Next, we perform the multiplication:

2 * 6.2 = 12.4

Substitute this back into the equation:

  1. 4 - 7.6 = 4.8

Finally, we perform the subtraction:

  1. 4 - 7.6 = 4.8

So, we have 4.8 = 4.8, which is a true statement! This confirms that our solution, j = 18.5, is correct. Checking your work is a fantastic habit to get into because it helps you catch any errors and build confidence in your problem-solving skills.

Conclusion

Great job, everyone! We've successfully solved for j in the equation 2(j-12.3)-7.6=4.8. We found that j = 18.5. Remember, the key to solving algebraic equations is to isolate the variable by using inverse operations and keeping the equation balanced. By following these steps – distributing, combining like terms, adding or subtracting from both sides, and dividing or multiplying both sides – you can tackle a wide range of algebraic problems.

Don't forget to always check your solution by plugging it back into the original equation. This ensures that your answer is correct and helps you build confidence in your skills.

Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to try another problem, feel free to ask. Happy solving!