Algebra
Linear Equations
One-Step Equations
Solve equations requiring a single operation like or .
A one-step equation requires exactly one inverse operation to isolate the variable. For example:
- : subtract from both sides →
- : divide both sides by →
- : add to both sides →
- : multiply both sides by →
The key is identifying which operation is being applied to the variable and using its inverse.
Multi-Step Equations
Solve equations requiring multiple operations like .
Multi-step equations require more than one inverse operation. The strategy is to reverse the order of operations (undo addition/subtraction first, then multiplication/division):
For :
- Add to both sides:
- Divide both sides by :
Always check by substituting back into the original equation.
Equations with Fractions
Solve equations involving fractions like .
When solving equations with fractions, you have two approaches:
- Clear the fraction: multiply both sides by the denominator
- Treat it as a multi-step: undo addition/subtraction first, then the fraction (which is division by the denominator)
For :
- Subtract :
- Multiply by :
For equations with multiple fractions, multiplying by the least common denominator (LCD) clears all fractions at once.
Variables on Both Sides
Solve equations with variables on both sides like .
When the variable appears on both sides, the goal is to collect variable terms on one side and constant terms on the other:
For :
- Subtract from both sides:
- Subtract from both sides:
Tip: always move the smaller variable term to avoid negative coefficients when possible.
Concepts
What is a Linear Equation?
A linear equation is an equation where the highest power of the variable is . The general form is , where is the variable, and , , are constants with . The graph of a linear equation in one variable is a single point on the number line; in two variables, it forms a straight line.
The Balance Principle
Solving equations relies on the balance principle: whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This preserves equality. Think of a balanced scale — if you add weight to one side, you must add the same weight to the other to keep it balanced.
Inverse Operations
To isolate the variable, use inverse operations:
| Operation | Inverse Operation |
|---|---|
| Addition () | Subtraction () |
| Subtraction () | Addition () |
| Multiplication () | Division () |
| Division () | Multiplication () |
| Square () | Square Root () |
The goal is to undo everything that has been done to the variable, in reverse order (reverse of PEMDAS).
Checking Solutions
Always verify your solution by substituting it back into the original equation. If both sides evaluate to the same value, the solution is correct. For example, if solves , then confirms it.
Worked Examples
One-Step Equation
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Solution Steps
Answer
Two-Step Equation
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Solution Steps
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Variables on Both Sides
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Solution Steps
Answer
Equation with Fractions
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Solution Steps
Answer