Mastering Systems of Equations: A Step-by-Step Guide to Finding x

Algebra can sometimes feel like a puzzle, can’t it? You work through it, piece by piece, and just when you think you’ve got it, another challenge pops up. Today, we’re diving into a commonly faced aspect of algebra: solving systems of equations. Let’s walk through this together. Ready? Let’s get started!

Here’s the situation we’re facing: We need to find the value of \(x\) in a system of equations given as follows:

1. \(3x - 3y = 3\) 
2. \(x - 5y = -3\)

At first glance, it may seem a little overwhelming, but hang in there. We’ll simplify and solve these equations step-by-step. 

**Step 1: Simplifying the First Equation**

Our first equation is \(3x - 3y = 3\). What we can do is simplify this by dividing everything by 3. When we do that, we get:

\[
x - y = 1 \quad \text{(Equation 1)}
\]

Yeah, that’s much cleaner, right? 

**Step 2: Rearranging the First Equation**

Now, from Equation 1, we can express \(x\) in terms of \(y\):

\[
x = y + 1
\]

**Step 3: Substituting into the Second Equation**

Next, we’ll substitute that expression for \(x\) into the second equation, which is \(x - 5y = -3\). So, placing the expression we found into Equation 2 gives us:

\[
(y + 1) - 5y = -3
\]

**Step 4: Combining Like Terms**

Now let’s combine like terms. We have:

\[
y + 1 - 5y = -3
\]

Simplifying that, we end up with:

\[
-4y + 1 = -3
\]

Get your thinking caps on because we're just about to finish this part!

**Step 5: Isolating y**

Let’s subtract 1 from both sides to isolate the term with \(y\):

\[
-4y = -4
\]

Now, dividing both sides by -4 results in:

\[
y = 1
\]

Great! We found \(y\). But remember, folks, we’re not done just yet.

**Step 6: Finding x**

Now that we have \(y\), we can find \(x\) using the equation \(x = y + 1\). Substituting \(y = 1\) gives us:

\[
x = 1 + 1 = 2
\]

**Conclusion: Value of x**

So, the value of \(x\) in our system of equations is 2. Feeling ready to take on similar problems now? 

Solving systems of equations is like learning how to ride a bike— at first, it’s wobbly, but with practice, you’ll find your balance. Keep solving, keep questioning. And remember, each equation you conquer keeps you one step closer to algebra mastery. And hey, don't hesitate to revisit these concepts as they pop up in your studies!

Happy solving!