Solve with variables on both sides
When a linear equation has variables on both sides, the strategy is to collect all variable terms on one side and all constants on the other, then solve using inverse operations. For example, to solve 3x + 5 = x + 13, subtract x from both sides to get 2x + 5 = 13, then subtract 5 to get 2x = 8, and divide to find x = 4. This is a fundamental skill in Chapter 2 of OpenStax Elementary Algebra 2E. Students who master this technique can handle any linear equation regardless of how the terms are distributed.
Key Concepts
Property For equations with variables on both sides of the equation, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.
Examples Solve $10x = 9x 7$. Subtract $9x$ from both sides to get all the variables on the left, which gives the solution $x = 7$.
Solve $3y 8 = 6y$. Subtract $3y$ from both sides to gather the variables on the right, which gives $ 8 = 3y$. Divide by 3 to get $y = \frac{8}{3}$.
Common Questions
How do you solve a linear equation with variables on both sides?
Move all variable terms to one side by adding or subtracting, then move all constants to the other side. Combine like terms and divide by the coefficient to isolate the variable.
What is the first step when variables appear on both sides?
Choose one side to collect variables on, then use addition or subtraction to move the variable term from the other side. It does not matter which side you choose, but picking the side with the larger coefficient avoids negatives.
How do I know if my solution is correct?
Substitute your answer back into the original equation and verify both sides are equal. If they are, the solution is correct.
What does it mean if both variables cancel and you get a false statement?
If variables cancel and you get something like 3 = 7, the equation has no solution — it is a contradiction.
What if variables cancel and you get a true statement?
If you get something like 0 = 0, the equation has infinitely many solutions — it is an identity, and every real number satisfies it.
When do students learn to solve equations with variables on both sides?
This is a core algebra 1 skill, covered in OpenStax Elementary Algebra 2E Chapter 2: Solving Linear Equations and Inequalities.
What is a common mistake when solving equations with variables on both sides?
Not applying the same operation to every term on both sides, or forgetting to distribute when one side has parentheses.