Does it matter which side I move the variables to?

No, it doesn't matter at all. Whether you get $x = 4$ or $4 = x$, the meaning is exactly the same. However, moving the smaller variable term toward the larger one usually prevents you from having to divide by a negative number at the end.

What happens if all the variables cancel out?

If your variables cancel out completely, you'll be left with a statement comparing two numbers. If the statement is true (like $5 = 5$), the equation has infinite solutions (it's an identity). If the statement is false (like $5 = 8$), the equation has no solution.

Can I move the numbers before I move the variables?

Absolutely. The order in which you move the terms doesn't change the final answer, as long as you follow the rules of algebra and do the same operation to both sides. Many students find it easier to group variables first, but either way works.

How Do You Solve a Linear Equation With Variables on Both Sides?

To solve a linear equation with variables on both sides, your main goal is to gather all the terms containing the variable onto one side of the equal sign, and all the plain numbers (constants) onto the other. Once they are separated, you can isolate the variable just like you would in a standard two-step equation.

Think of the equation like a perfectly balanced scale with a mix of mystery boxes (the variables) and standard weights (the numbers) on both pans. To figure out the weight of one mystery box, you need to physically remove boxes from one side and weights from the other, always doing the exact same thing to both sides so the scale never tips. In algebra, we do this using inverse operations like addition and subtraction.

Step 1: Gather the Variables

The first step is to pick one side for your variable. It doesn't mathematically matter which side you choose, but a good trick is to move the smaller variable term toward the larger one. This helps you avoid dealing with negative numbers later. To move a term, you use the opposite operation. If you have a positive $2x$ on the right, you subtract $2x$ from both sides.

Step 2: Gather the Constants

Once all your variables are on one side, you need to get the plain numbers to the opposite side. If your variable is on the left, move any numbers on the left over to the right. Again, you do this by adding or subtracting the exact same amount from both sides of the equals sign.

Step 3: Isolate the Variable

After sorting the terms, you will usually be left with something like $3x = 12$ . The final step is to undo the multiplication or division attached to your variable. If the variable is multiplied by 3, you divide both sides by 3 to find out what a single $x$ equals.

Where Students Slip Up

The most common mistake is "moving" a term across the equal sign without actually changing its sign. For example, if a student wants to move $+4x$ to the other side, they might accidentally add $4x$ instead of subtracting it. Always write out your steps and double-check your positive and negative signs.

Worked through

Solve the equation for $x$ : $5x - 3 = 2x + 9$

First, we want to get the variables on one side. Let's move the $2x$ from the right side to the left side. Since it's a positive $2x$ , we subtract $2x$ from both sides: $5x - 2x - 3 = 2x - 2x + 9$ $3x - 3 = 9$

Next, we need to move the constant ( $-3$ ) to the right side. The opposite of subtracting 3 is adding 3, so we add 3 to both sides: $3x - 3 + 3 = 9 + 3$ $3x = 12$

Finally, to get $x$ by itself, we divide both sides by 3: $\frac{3x}{3} = \frac{12}{3}$ $x = 4$

You can check this by plugging $4$ back into the original equation: $5(4) - 3 = 17$ , and $2(4) + 9 = 17$ . It balances!

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Where this comes from: OpenStax Elementary Algebra, Chapter 2: Solving Linear Equations · Khan Academy, Algebra 1: Solving equations with one unknown