Solving linear equations can be tricky sometimes, especially when you see variables on both sides. But once you get the hang of it, it becomes much easier. The main goal is to isolate the variable you want to solve for, usually shown as $x$.

The Basics of Solving Equations

Let’s start with a simple equation:

$$2x + 3 = 11$$

In this case, we can find $x$ by moving the constant (which is the number 3 here) from the left side to the right side. We do this by subtracting 3 from both sides:

$$2x = 11 - 3$$

Now, let’s simplify that:

$$2x = 8$$

Finally, we divide both sides by $2$ to find:

$$x = 4$$

This was easy because there was no variable on the right side.

When Variables are on Both Sides

Things get a bit interesting when we have variables on both sides. For example:

$$3x + 5 = 2x + 9$$

Now, we need to figure out how to isolate $x$.

Step 1: Identify the Variables

First, look closely at where the variables are. Here, we have $3x$ on the left and $2x$ on the right. To solve for $x$, we can move one of these terms to one side.

Step 2: Move the Variables

Let’s simplify by subtracting $2x$ from both sides:

$$3x - 2x + 5 = 9$$

This simplifies to:

$$x + 5 = 9$$

Step 3: Isolate the Variable

To get $x$ by itself, we can move $5$ to the other side. We do this by subtracting $5$:

$$x = 9 - 5$$

So we find:

$$x = 4$$

When to Move Variables

Now, how do we know when to move variables? Here are some tips:

1. Look for Variables and Constants: Always check where your variables and numbers are in the equation.

2. Pick One Side for All Variables: Decide if you want all your variables on the left side or the right. It doesn’t matter which side, just be consistent.

3. Use Operations to Isolate: Use addition or subtraction to move the variables and constants to their sides so you can isolate the variable.

4. Keep the Equation Balanced: Remember, if you change one side of the equation, you must do the same to the other side. This keeps everything equal.

Practice Makes Perfect

To understand these ideas better, practicing with different equations helps. Here’s another example to try:

$$4x - 2 = 6x + 8$$

Give it a go on your own using the steps we talked about. The more you practice, the more comfortable you will feel moving variables around!

In conclusion, when solving linear equations with variables on both sides, remember these rules. The more you practice, the easier it will become!