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\(x\)
\(x\)
\(=\)

Equations with Variables
on Both Sides

Learn to solve equations like \( 3x + 5 = x + 13 \) by collecting variables on one side and constants on the other.

Pre-Algebra

What Does "Variables on Both Sides" Mean?

So far, most equations you have solved look like \( 2x + 3 = 11 \): the variable \( x \) appears on only one side. Now we level up.

An equation with variables on both sides has the unknown on the left and the right of the equals sign:

Our goal is still the same: find the value of \( x \) that makes both sides equal.

Think of a Balance Beam

\(5x + 2\)
\(3x + 10\)
\(=\)
 Both sides must stay balanced!

Why Does This Matter?

Real Life
Comparing phone plans, budgets, or speeds often gives equations with variables on both sides.
Foundation
This skill is the gateway to solving systems of equations, inequalities, and higher algebra.
Flexibility
You will see these in word problems, geometry, and science formulas throughout middle school and beyond.

The Core Strategy

Original
\(5x + 2 = 3x + 10\)
 Subtract 3x
\(2x + 2 = 10\)
 Subtract 2
\(2x = 8\)
 Divide by 2
\(x = 4\)

The Formal Rule

You do not need to memorize this formula. Instead, understand the process: move variables to one side, constants to the other, then divide. The formula simply describes what happens when you do those steps.

One Side vs Both Sides

Variable on One Side Variables on Both Sides
Example \( 2x + 5 = 11 \) \( 5x + 2 = 3x + 10 \)
First Step Undo the constant Move a variable term
Key Move Subtract 5 from both sides Subtract \( 3x \) from both sides
Then Divide to isolate \( x \) Now solve like a one-side equation

How to Solve: Step by Step

1
Simplify each side if needed (distribute, combine like terms).
2
Move variable terms to one side by adding or subtracting.
3
Move constant terms to the other side by adding or subtracting.
4
Divide (or multiply) to isolate the variable.
5
Check your answer by plugging it back into the original equation.

Example 1: Basic

1
Subtract \( 2x \) from both sides: \( 4x - 2x + 3 = 9 \) → \( 2x + 3 = 9 \)
2
Subtract \( 3 \) from both sides: \( 2x = 6 \)
3
Divide both sides by \( 2 \): \( x = 3 \)

Example 2: Negatives

1
Subtract \( 2x \) from both sides: \( 3x + 25 = 1 \)
2
Subtract \( 25 \) from both sides: \( 3x = -24 \)
3
Divide both sides by \( 3 \): \( x = -8 \)

Example 3: Distribute First

1
Distribute: \( 3x + 12 = 5x - 2 \)
2
Subtract \( 3x \) from both sides: \( 12 = 2x - 2 \)
3
Add \( 2 \) to both sides: \( 14 = 2x \)
4
Divide by \( 2 \): \( x = 7 \)

Special Cases

Identity (All Solutions)
\( 2x + 6 = 2(x + 3) \) simplifies to \( 6 = 6 \). This is always true, so every value of \( x \) works.
No Solution
\( 3x + 1 = 3x + 5 \) simplifies to \( 1 = 5 \). This is never true, so there is no value of \( x \) that works.

Visualizing the Solution

Where do the two sides of \( 4x + 3 = 2x + 9 \) become equal? Graph both as lines and find the intersection.

\(x\)
\(y\)
 0 1 2 3 4 5 10 15 20
\(y = 4x + 3\)
\(y = 2x + 9\)
\((3, 15)\)

Move Left or Right?

You can collect variables on either side. A good strategy: move the smaller variable term to the side of the larger one. This avoids negative coefficients.

✓ Smart Move
\( 2x + 5 = 6x - 3 \)
Subtract \( 2x \): \( 5 = 4x - 3 \)
Positive coefficient!
✕ Harder Move
\( 2x + 5 = 6x - 3 \)
Subtract \( 6x \): \( -4x + 5 = -3 \)
Negative coefficient.

Common Mistakes

✕ Forgetting to Distribute
In \( 2(x + 3) = \ldots \), students write \( 2x + 3 \) instead of \( 2x + 6 \). Always multiply every term inside the parentheses.
✕ Sign Errors
Subtracting a negative is adding. If you subtract \( -3x \) from both sides, you are really adding \( 3x \).
✕ Moving Without Balancing
Students "move" a term to the other side without performing the operation on both sides. The equation becomes unbalanced.

Summary Checklist

Step Action Example
1 Simplify each side \( 3(x+2) \to 3x+6 \)
2 Move variable terms Subtract \( 2x \) from both sides
3 Move constants Add or subtract a number
4 Isolate variable Divide both sides
5 Check Plug \( x \) into original

Question 1 of 5

Solve: \( 6x + 1 = 4x + 9 \)

Question 2 of 5

Solve: \( 5x - 3 = 2x + 12 \)

Question 3 of 5

Solve: \( 2(x + 5) = x + 16 \)

Question 4 of 5

What happens when you solve \( 4x + 7 = 4x + 7 \)?

Question 5 of 5

Solve: \( 8 - 3x = 2x - 7 \)