What if all variable terms cancel out and leave a false statement?

If you obtain a statement like $4 = 7$, the equation has no solution. It is a contradiction.

What if all variable terms cancel out and leave a true statement?

If you obtain a statement like $5 = 5$, the equation has infinitely many solutions. It is an identity.

Does it matter which side I move the variables to?

No. Moving variables to either side yields the exact same result. Moving them to the side with the larger coefficient avoids negative coefficients.

How to solve an equation with variables on both sides

To solve an equation with variables on both sides, apply inverse operations to collect all variable terms on one side and all constant terms on the other. This method applies to linear equations where the unknown appears in multiple terms across the equals sign. Once the variables are grouped and simplified into a single term, isolate the variable by dividing or multiplying by its coefficient. The resulting statement provides the exact value of the unknown.

The setup

Identify the equation format $ax + b = cx + d$ . Verify that both sides are simplified. If there are parentheses, distribute first. Combine any like terms on each individual side before moving terms across the equals sign.

The steps

Choose one side for the variable terms and the other for constants. 2. Add or subtract the variable term from one side to move it to the chosen side. 3. Add or subtract the constant term from the variable's side to move it to the constant side. 4. Simplify both sides to the form $kx = C$ . 5. Divide both sides by $k$ to isolate $x$ .

Checking the result

Substitute the calculated value of $x$ back into the original equation. Evaluate the left side and the right side independently. If both sides evaluate to the identical numerical value, the solution is correct.

Common errors

Failing to apply the operation to both sides simultaneously. Dropping a negative sign when subtracting a larger coefficient from a smaller one. Adding terms that are not like terms, such as combining an $x$ term with a constant.

Worked example

Solve for $x$ : $3(x - 4) = 5x + 8$

First, distribute the 3 on the left side: $3x - 12 = 5x + 8$ . Next, subtract $3x$ from both sides to collect variables on the right: $-12 = 2x + 8$ . Subtract 8 from both sides to collect constants on the left: $-20 = 2x$ . Finally, divide by 2: $-10 = x$ . Therefore, $x = -10$ .

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References: OpenStax Elementary Algebra, Chapter 2: Solving Linear Equations · Khan Academy: Solving equations with variables on both sides