Math · Algebra & word problems
Solving Linear Equations
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Start here: key ideas
- An equation states that two expressions are equal.
- Use the same valid operation on both sides.
- Simplify each side before isolating the variable.
- Substitution checks the original equation.
What you’ll be able to do
- Preserve equality with inverse operations.
- Solve one-step and multi-step equations.
- Verify by substitution.
Treat an equation as a balance: change both sides equally until the variable stands alone.
Complete Expressions and Order of Operations first.
Equality as a balance
An equation states that two expressions have equal value. A solution makes that statement true. Equality behaves like a balanced scale: adding, subtracting, multiplying, or dividing both sides by the same nonzero value preserves balance. Inverse operations undo one another, such as +7 and −7.
Isolate the variable
To isolate a variable, reverse operations in an order that keeps both sides equal. For 3x+5=20, subtract 5 from both sides to get 3x=15, then divide both sides by 3 to get x=5. Write one equality-preserving step per line.
| Reason | Left side | Right side |
|---|---|---|
| Start | 3x+5 | 20 |
| Subtract 5 | 3x | 15 |
| Divide by 3 | x | 5 |
Multi-step equations and checking
Use the distributive property and combine like terms first. Variables may appear on both sides; move one variable term so the remaining coefficient is convenient. Never “move” a term without naming the inverse operation applied to both sides.
Substitution checks a solution by replacing the variable in the original equation. If x=5 in 3x+5=20, then 3(5)+5=20, a true statement. A false statement means the proposed value or prior algebra is wrong.
Worked examples
Worked example 1: 4(x−2)+3=19. Distribute: 4x−8+3=19. Combine: 4x−5=19. Add 5: 4x=24. Divide by 4: x=6. Check: 4(6−2)+3=19.
Worked example 2: 7x+2=3x+18. Subtract 3x: 4x+2=18. Subtract 2: 4x=16. Divide: x=4. Both original sides equal 30.
A reliable way to reason
Solving is a sequence of equivalent equations, not a process of moving symbols. In 5x−8=27, adding 8 to both sides creates 5x=35 because equal amounts were added to equal quantities. Dividing both sides by 5 creates x=7. Writing the operation beside each line makes sign changes explainable and keeps the balance visible.
Choose steps that reduce complexity. Distribute and combine like terms within each side before moving variable terms. For 3(x+2)−x=14, distribution gives 3x+6−x=14, then 2x+6=14, then x=4. If variables occur on both sides, subtract one variable term from both sides; choosing the smaller coefficient often leaves a positive coefficient. Special results have meaning. If variables cancel and a true statement remains, every real number works. If a contradiction remains, no value works. For x+2=x+2, subtracting x leaves 2=2, so every x works. For x+2=x+3, it leaves 2=3, so no x works. Substitution is decisive: place the proposed value into the original equation, evaluate both sides independently, and confirm equality. This catches distribution and sign errors that may appear consistent in later work.
Guided reasoning and error checks
Fractions and decimals do not change the balance principle. For x/3+1/2=5/2, subtract 1/2 from both sides to get x/3=2, then multiply both sides by 3: x=6. Alternatively, multiply the original equation by 6 to clear denominators: 2x+3=15, then 2x=12 and x=6. Both routes preserve equality.
For 0.4x−1.2=2.8, add 1.2 to obtain 0.4x=4.0, then divide by 0.4 to obtain x=10. Substitute: 0.4(10)−1.2=2.8. The decimal estimate also agrees: four tenths of ten is four, slightly reduced by 1.2. Using both exact substitution and a magnitude check gives independent evidence.
Final self-check
If a solution is fractional, keep it exact unless the problem requests rounding. From 6x+1=4, x=1/2. Substitution gives 6(1/2)+1=4 exactly. Writing 0.5 is also exact here, but a repeating decimal such as 1/3 is safer as a fraction because early rounding may make the check appear unequal.
Key terms
- equation: A statement that two expressions are equal.
- solution: A value that makes an equation true.
- inverse operations: Operations that undo each other.
- isolate: Leave a variable alone on one side.
- substitution: Replacing a variable with a value.
- distributive property: Multiplying a factor by every term in a group.
Summary
- Every operation must preserve the balance.
- Distribute and combine like terms before isolating.
- Check in the original equation, not a possibly flawed intermediate line.
- Substitute the proposed solution into the original equation and verify that both sides are equal.
Simplify, undo addition/subtraction, undo multiplication/division, then substitute to check.
Sources
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