Algebra basics
Treat variables as numbers whose value may be unknown and use inverse operations to isolate them.
Expressions and equations#
An expression such as 3x + 5 represents a value. An equation such as 3x + 5 = 20 states that two expressions are equal.
Like terms have matching variable parts. 3x + 2x becomes 5x, but 3x + 2 does not.
Preserve equality#
An equation behaves like a balanced scale. Apply the same valid operation to both sides.
Solve 3x + 5 = 20:
- subtract 5 from both sides: 3x = 15;
- divide both sides by 3: x = 5.
Check: 3(5) + 5 = 20.
Distribute carefully#
The distributive property gives a(b + c) = ab + ac. Therefore 2(x + 4) = 2x + 8. Multiply every term inside the bracket.
Translate words#
“Five more than twice a number” becomes 2x + 5. Order matters: “five less than a number” is x - 5, not 5 - x.
Common mistakes#
Do not move a term across the equals sign by changing its sign without understanding that you are applying an inverse operation to both sides. Avoid combining unlike terms, and remember that dividing by zero is never valid.
Final check#
Substitution is the quickest verification. If the solution does not make both sides equal, revisit arithmetic, signs, and distribution.
Key points
- A variable represents a number that may be unknown or allowed to change.
- Like terms can be combined only when their variable parts match.
- Apply the same valid operation to both sides of an equation.
- Substitute the result into the original equation to check it.