How to Solve Equations Both Sides

To solve equations both sides, keep balance: simplify each side, use inverse operations to move terms, and isolate the variable step by step.

If you came here to learn how to solve equations both sides, you’re in the right place. The equal sign means both sides carry the same value, like a scale. Every move you make to one side must happen on the other. That balance idea powers every method below and keeps your solution trustworthy.

How To Solve Equations Both Sides: Core Steps

Here’s the reliable playbook for linear equations with a single variable on both sides. You’ll simplify, group like terms, isolate the variable, and check your answer. The moves rest on the properties of equality and inverse operations taught in standard algebra courses and open textbooks, so the method works across problems ranging from friendly integers to messy fractions (see the Subtraction and Addition Properties of Equality and OpenStax key concepts page for the formal statements).

Simplify each side. Distribute, combine like terms, and clean up fractions or decimals if needed.
Group variable terms. Add or subtract to get all variable terms on one side and constants on the other.
Isolate the variable. Use inverse operations (add/subtract, multiply/divide) to get the variable alone.
Check. Plug the value back into the original equation to confirm both sides match.

Broad Moves You’ll Use Early And Often

These are the standard “legal” actions that keep the equation balanced and move you toward the solution. The first table appears early so you can scan the options while you work.

Move	What It Does	Quick Example
Add The Same Value	Shifts both sides equally; used to clear negatives or group terms.	From x − 7 = 5 to x = 12
Subtract The Same Value	Removes a constant from one side; helps isolate the variable.	From 3x + 9 = 24 to 3x = 15
Multiply Both Sides	Clears denominators or scales; don’t use zero.	From x/5 = 3 to x = 15
Divide Both Sides	Undoes multiplication; don’t divide by zero.	From 4x = 28 to x = 7
Distribute	Removes parentheses: a(b + c) = ab + ac.	From 2(x + 3) to 2x + 6
Combine Like Terms	Condenses expressions so you can see the path to isolate.	From 5x − 2x to 3x
Clear Fractions	Multiply by the least common denominator to work with integers.	From (x/3) + (1/6) to multiply through by 6
Substitute To Check	Verifies the solution by plugging the value back.	If x = 7, check 4x = 28 → true

Formal statements of these properties—and lots of examples—appear in free algebra resources. You can read the addition, subtraction, multiplication, and division properties of equality in the OpenStax concept summary, then see hands-on practice sets for variables on both sides to build speed.

Worked Examples: From Clean To Messy

Example 1: Integers, No Parentheses

Solve: 7x − 10 = 3x + 22

Group variable terms: subtract 3x on both sides → 4x − 10 = 22.
Move constants: add 10 on both sides → 4x = 32.
Isolate: divide both sides by 4 → x = 8.
Check: LHS = 7(8) − 10 = 46; RHS = 3(8) + 22 = 46 → match.

Example 2: Parentheses And Distribution

Solve: 3(x − 4) + 8 = 2x + 5

Distribute: 3x − 12 + 8 = 2x + 5 → 3x − 4 = 2x + 5.
Group variable terms: subtract 2x → x − 4 = 5.
Isolate: add 4 → x = 9.
Check: LHS = 3(9 − 4) + 8 = 3(5) + 8 = 23; RHS = 2(9) + 5 = 23.

Distribution step follows the standard property a(b + c) = ab + ac; many textbooks present it as a first move when parentheses block isolation. See a concise definition and practice under the distributive property in open resources for algebra learners.

Example 3: Fractions On Both Sides

Solve: (x/4) − (3/2) = (x/6) + 1

Clear fractions: the LCD of 4, 2, and 6 is 12. Multiply both sides by 12 → 3x − 18 = 2x + 12.
Group variable terms: subtract 2x → x − 18 = 12.
Isolate: add 18 → x = 30.
Check in the original fractional form: LHS = 30/4 − 3/2 = 7.5 − 1.5 = 6; RHS = 30/6 + 1 = 5 + 1 = 6 → match.

Clearing denominators is a classic speed move when you’re solving with fractions; multiplying both sides by the least common denominator preserves equality and removes the clutter so the inverse steps are easier to see.

Example 4: Decimals And Combining Like Terms

Solve: 0.4x + 1.8 = 1.1x − 2.5

Group variable terms: subtract 0.4x → 1.8 = 0.7x − 2.5.
Move constants: add 2.5 → 4.3 = 0.7x.
Isolate: divide by 0.7 → x = 4.3/0.7 = 6.142857…
Check with rounding awareness: plug back to confirm both sides match to your rounding level.

Example 5: Many Steps, Mixed Operations

Solve: 5 − 2(3x − 1) = 4x + 17

Distribute: 5 − 6x + 2 = 4x + 17 → 7 − 6x = 4x + 17.
Group variable terms: add 6x → 7 = 10x + 17.
Move constants: subtract 17 → −10 = 10x.
Isolate: divide by 10 → x = −1.
Check to confirm both sides equal 27 when x = −1.

Why These Steps Work

The method comes from the properties of equality: if two expressions are equal, doing the same legal operation to both sides keeps them equal. That includes adding or subtracting the same value, and multiplying or dividing both sides by the same nonzero value. Those formal properties are stated in plain language with examples in the OpenStax summary of algebraic equality rules, and you can watch short problem walk-throughs for equations with variables on both sides in free video lessons by trusted math educators.

Common Pitfalls And How To Avoid Them

Even with a clean plan, small slips can derail a solution. This section shows what to watch for and the quick fix each time.

Forgetting To Distribute

Slip: Treating 2(x + 5) as 2x + 5. Fix: Multiply through every term inside the parentheses → 2x + 10.

Moving Terms To The Wrong Side

Slip: Subtracting 3x from both sides of 5x − 7 = 3x + 11 but writing 5x − 7 − 3x = 3x + 11 (and leaving 3x on the right). Fix: Write both sides after the move: (5x − 3x) − 7 = 11 → 2x − 7 = 11.

Dropping Signs

Slip: Turning −4 into +4 when moving across the equals sign. Fix: Think “add the opposite” on both sides, not “move across.”

Mixing Unlike Terms

Slip: Combining 2x + 5 as 7x. Fix: Only combine x-terms with x-terms and constants with constants.

Skipping The Check

Slip: Stopping after isolation. Fix: Substitute back into the original version. A 10-second check saves points.

Handling Fractions, Parentheses, And Negatives

Trickier problems mostly come from extra structure on each side. The cure is the same: simplify the structure, then isolate.

Clear Denominators Early

Multiply both sides by the least common denominator so you can work with integers. Keep track of every term; the scale stays balanced only if you multiply the full side, not just a part. Many step-by-step lessons model this move clearly for “variables on both sides,” including how to confirm the solution once the fractions are gone.

Distribute, Then Combine

When parentheses block isolation, distribute first, then combine like terms. This is faster than moving terms before you expand.

Watch Negative Coefficients

If a variable carries a negative coefficient, you can multiply both sides by −1 to tidy the expression before you isolate. It’s a legal balance move and reduces sign slips later.

Special Results: No Solution And Infinitely Many Solutions

Not every equation leads to a single number. During isolation you may land on a true statement with no variable (like 0 = 0) or a false statement (like 4 = 7). Here’s what those outcomes mean.

Identity: All Real Numbers

Example: 2(x + 3) − 4 = 2x + 2

Distribute and combine: 2x + 6 − 4 = 2x + 2 → 2x + 2 = 2x + 2.
Subtract 2x on both sides: 2 = 2 → always true.

Conclusion: Every real number solves the equation. Any x you plug in keeps both sides equal because the two sides were algebraically the same expression.

Contradiction: No Solution

Example: 3(x − 1) + 5 = 3x − 1

Distribute: 3x − 3 + 5 = 3x − 1 → 3x + 2 = 3x − 1.
Subtract 3x: 2 = −1 → false.

Conclusion: No value of x makes the original equation true. The expressions never match.

Translating Word Problems To “Both Sides” Equations

Many real-world statements become equations with a variable on each side. The trick is to name a variable, write each side as an expression, then set them equal because the two quantities match in the story.

Steps For Translation

Name the variable. Let x be the unknown quantity the question asks for.
Build each side. Write the cost, distance, or count expression for each side of the scenario.
Set them equal. Use the equal sign when the story states two expressions match.
Solve and check. Use the core steps, then read the question again to report the unit.

Quick Model

Story: A streaming plan costs a base fee plus a per-movie charge. Another provider charges a different base fee and a smaller per-movie charge. At how many movies do the plans cost the same?

Set up: x = number of movies. Plan A: 199 + 40x. Plan B: 299 + 30x. Set equal: 199 + 40x = 299 + 30x → 10x = 100 → x = 10. At 10 movies, the costs match.

Practice Plan That Builds Speed

Accuracy comes first, then speed. Start with integer-based equations, add parentheses, then move to fractions and decimals. Mix in identities and contradictions so you can spot them early. For extra reps, run problem sets where the variable shows up on both sides; many free platforms offer graded practice and quick feedback while you work through your steps.

Troubleshooting: Mistakes And Quick Fixes

Common Mistake	Why It Happens	Quick Fix
Combining Unlike Terms	Forgetting x-terms and constants are different kinds.	Group x-terms first, then constants. Rewrite lines cleanly.
Half-Distributing	Multiplying only the first term inside parentheses.	Draw arrows to each term; say “hit every term.”
Fraction Fatigue	Working term-by-term without clearing denominators.	Multiply both sides by the LCD at the start.
Sign Slips	Dropping negatives during moves or combination.	Use parentheses when adding the opposite; slow one line.
Stopping Before Check	Time pressure or overconfidence.	Plug the answer back; it takes seconds and catches slips.
Reporting The Wrong Quantity	Solving x but the question asked for another value.	Read the last sentence again before you write the final line.
Dividing By Zero	Trying to divide by a coefficient that became 0.	If the coefficient is 0, look for identity/contradiction cases.

Checks That Keep You Honest

Two checks give you confidence: substitution and units/context.

Substitution: Place your x-value back into the original equation—not the simplified one. Both sides should match.
Units/context: In story problems, the answer should fit real-world units and make sense. Negative length or a fee that can’t be negative is a red flag.

One-Page Template You Can Follow Every Time

Write the equation clear. Copy it carefully so signs and terms are correct.
Simplify each side. Distribute and combine like terms.
Group variables on one side. Add/subtract to move x-terms together.
Move constants to the other side. Keep the variable’s side clean.
Isolate the variable. Multiply or divide to get x by itself.
Check. Substitute the value into the original equation.
State the answer cleanly. Include units if it’s a word problem.

Where To Practice And Read More

If you want extra drill on how to solve equations both sides, try practice sets that give instant feedback and see a structured write-up of the equality rules in a free textbook. A balanced routine—five careful problems a day—beats long sessions that lead to slips. Here are two starter spots you can keep open in a tab while you work: variables on both sides practice and a concise list of equality properties in the OpenStax key concepts.

Recap You Can Memorize

Balance the equation, simplify each side, gather like terms, isolate with inverse operations, and check. That’s the loop. With these steps and the two linked references above, you’ll be ready for any one-variable linear equation where the variable shows up on both sides.