Solving The Linear Equation: $\frac{1}{2}-x+\frac{3}{2}=x-4$

Hey guys! Today, we're going to dive into solving a linear equation. Linear equations are a fundamental part of algebra, and mastering them is super important for tackling more complex math problems later on. We're going to take a step-by-step approach to solve the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$. Don't worry, it's not as scary as it looks! We'll break it down together and make sure you understand each part. So, grab your pencils and let's get started!

Understanding Linear Equations

First, let's quickly recap what a linear equation actually is. At its core, a linear equation is an algebraic equation where the highest power of the variable (in our case, x) is 1. This means you won't see any $x^2$, $x^3$, or other higher powers. Linear equations, when graphed, produce a straight line – hence the name! Think of it as a balanced scale: our goal is to keep the equation balanced while isolating the variable x on one side. To do this, we use inverse operations, which are operations that "undo" each other (like addition and subtraction, or multiplication and division).

Think of it like this: you have a puzzle, and the x is the missing piece. Our job is to figure out what that piece is by carefully manipulating the puzzle until we have x all by itself on one side of the equation. We achieve this by performing the same operations on both sides, ensuring that the equation remains balanced. Remember, whatever you do to one side, you must do to the other – that's the golden rule of equation solving! So, with this understanding in mind, let's jump into the first step of solving our equation.

Step 1: Combine Like Terms

The initial equation we're tackling is $\frac{1}{2}-x+\frac{3}{2}=x-4$. The first thing we want to do is simplify both sides of the equation as much as possible. This involves combining any like terms. Like terms are those that have the same variable raised to the same power (or are just constants). On the left side of our equation, we have two constant terms: $\frac{1}{2}$ and $\frac{3}{2}$. Let's combine them:

$\frac{1}{2} + \frac{3}{2} = \frac{1+3}{2} = \frac{4}{2} = 2$

So, the left side of the equation now simplifies to $2 - x$. Our equation now looks like this:

$2 - x = x - 4$

See? We've already made some progress! Combining like terms makes the equation cleaner and easier to work with. This step is all about tidying things up before we start moving terms around. By simplifying each side individually, we reduce the chances of making mistakes later on. Now that we've combined the constants on the left side, we're ready to move on to the next crucial step: getting all the x terms on one side of the equation.

Step 2: Move the x Terms to One Side

Now that we've simplified each side, our equation is $2 - x = x - 4$. Our next goal is to get all the x terms on the same side of the equation. It doesn't matter which side we choose, but it's often easier to move the x term that has a negative coefficient to the other side to avoid dealing with negative numbers later on. In this case, we have a $-x$ on the left side, so let's add x to both sides of the equation:

$2 - x + x = x - 4 + x$

This simplifies to:

$2 = 2x - 4$

By adding x to both sides, we've successfully eliminated the x term from the left side and combined it with the x term on the right side. This step is crucial because it brings us closer to isolating x. Remember, the key is to perform the same operation on both sides to maintain the balance of the equation. We're essentially "undoing" the subtraction of x on the left side by adding x. Now that we have all the x terms on one side, it's time to isolate them further by moving the constant terms to the other side.

Step 3: Move the Constants to the Other Side

Our equation is currently $2 = 2x - 4$. We want to isolate the term with x (which is $2x$), so we need to get rid of the $-4$ on the right side. To do this, we'll add 4 to both sides of the equation:

$2 + 4 = 2x - 4 + 4$

This simplifies to:

$6 = 2x$

Great! We've managed to get all the constant terms on the left side and the x term on the right side. By adding 4 to both sides, we've effectively "undone" the subtraction of 4. This step is another piece of the puzzle that brings us closer to finding the value of x. Now, we have a simple equation where x is multiplied by a coefficient. The final step involves isolating x by performing the inverse operation of multiplication, which is division.

Step 4: Isolate x by Dividing

We're now at the equation $6 = 2x$. To isolate x, we need to undo the multiplication by 2. We do this by dividing both sides of the equation by 2:

$\frac{6}{2} = \frac{2x}{2}$

This simplifies to:

$3 = x$

Or, we can write it as:

$x = 3$

Woohoo! We've done it! We've successfully solved for x. By dividing both sides by 2, we've isolated x and found its value. This final step is the culmination of all our previous efforts. We've combined like terms, moved variables and constants, and finally, we've isolated x. But before we celebrate too much, it's always a good idea to check our answer to make sure we haven't made any mistakes along the way.

Step 5: Check Your Answer

It's always a good idea to check your answer to make sure it's correct. To do this, we substitute our solution ($x = 3$) back into the original equation: $\frac{1}{2}-x+\frac{3}{2}=x-4$

Substitute $x = 3$:

$\frac{1}{2} - 3 + \frac{3}{2} = 3 - 4$

Now, let's simplify both sides. First, convert 3 into a fraction with a denominator of 2: $3 = \frac{6}{2}$. Then, we have:

$\frac{1}{2} - \frac{6}{2} + \frac{3}{2} = -1$

Combine the fractions on the left side:

$\frac{1 - 6 + 3}{2} = -1$

$\frac{-2}{2} = -1$

$-1 = -1$

Now, let's simplify the right side:

$3 - 4 = -1$

So, we have:

$-1 = -1$

Since both sides of the equation are equal, our solution $x = 3$ is correct! Checking our answer is a crucial step in problem-solving. It gives us confidence that our solution is accurate and helps us catch any mistakes we might have made along the way. By substituting our solution back into the original equation and verifying that both sides are equal, we can be sure that we've solved the equation correctly.

Conclusion

So, there you have it! We've successfully solved the linear equation $\frac{1}{2}-x+\frac{3}{2}=x-4$, and we found that $x = 3$. We tackled this problem step-by-step, from combining like terms to isolating the variable and finally, checking our answer. Remember, solving linear equations is all about keeping the equation balanced and using inverse operations to isolate the variable. With practice, you'll become a pro at solving these equations. Keep practicing, and you'll be able to solve even the trickiest linear equations with ease! Happy solving!