How do I solve a linear equation when the variable appears on both sides?
Quick answer
Simplify both sides first, then add or subtract the same terms on both sides to gather variable terms on one side and constants on the other. Isolate the variable and substitute any proposed value into the original equation to check it. The simplified result may give one solution, a true statement for every value, or a false statement with no solution. [S1][S2]
The question, the details, the next step.
I have an equation with x on both sides. What order should I use so I don’t lose track of the equals sign?
Keep the equation balanced: when you add or subtract a quantity on one side, do the same on the other. A reliable order is:
- Distribute through parentheses, if there are any.
- Combine like terms on each side.
- Gather variable terms on one side and constants on the other, using addition or subtraction on both sides.
- Isolate the variable, then check any proposed value in the original equation.
This is the general approach for solving linear equations described in OpenStax. [S1][S2] “Moving” a term across the equals sign is shorthand for applying the same operation to both sides; it does not mean changing only one side.
Can you walk me through an example where the variable appears on both sides and there are parentheses?
Consider 3(x + 2) − 4 = 2x + 9.
Distribute 3, then combine the constants on the left:
3x + 6 − 4 = 2x + 9
3x + 2 = 2x + 9
Subtract 2x from both sides:
x + 2 = 9
Subtract 2 from both sides:
x = 7
Now check in the original equation. Substitute 7 for x:
Left side: 3(7 + 2) − 4 = 27 − 4 = 23.
Right side: 2(7) + 9 = 14 + 9 = 23.
The sides match, so 7 satisfies this example. Checking the original equation tests the proposed answer against the equation you started with. [S2]
What if the variable terms cancel and I don’t end up with x equals a number?
Look at what remains after simplifying. If you get a true statement, such as 5 = 5, the equation is an identity: every value works. If you get a false statement, such as 5 = 8, the equation has no solution. If you instead get an equation you can solve for a particular value, it has a single solution. Collecting terms does not guarantee exactly one answer. [S1][S2]
For example, 2(x + 1) = 2x + 2 becomes 2x + 2 = 2x + 2. Subtract 2x from both sides to get 2 = 2. That is true for every value of x, rather than identifying one particular value.
Could you give me a small template I can reuse, including the check?
Use this outline and fill in each line with your working:
Original equation: [write both sides]
Simplify each side: [distribute and combine like terms]
Gather variable terms: [add or subtract the same term on both sides]
Gather constants and isolate the variable: [show each operation on both sides]
Classify the result: [one value / true statement for every value / false statement with no solution]
Check: Substitute [proposed value] into the original equation. The left side becomes [value], and the right side becomes [value]. For a proposed single solution, compare the two results. If simplification gives an identity or a contradiction instead, report that outcome rather than forcing an x-value. [S1][S2]
A practical checklist
Distribute through parentheses before combining terms.
Combine like terms separately on each side.
Use the same addition or subtraction on both sides to gather variable terms.
Isolate the variable, or identify whether the result is an identity or a contradiction.
For a proposed single value, substitute it into the original equation and compare both sides.
A few more useful answers
Does it matter which side I put the variable terms on?
Either side can work. Choose the side that makes the arithmetic easier to follow, and use the same operation on both sides.
If my check gives different values on the two sides, what should I do?
The proposed value has not passed the check. Review the distribution, signs, and operations in your working, then solve again and substitute the new result into the original equation.
Do I check an equation that simplifies to a true statement for every value?
There is no single proposed value to test. State that the equation is an identity and that every value satisfies it.
Sources
References used for this guide.
- [S1] OpenStax College Algebra with Corequisite Support: Linear Equations in One Variable openstax.org · Checked
- [S2] OpenStax Elementary Algebra: General Strategy for Solving Linear Equations openstax.org · Checked