Equations – The Language of Algebra
An equation is the mathematical equivalent of a balanced scale. Whatever you do to one side, you must do exactly the same to the other. This principle is the foundation of all equation solving.
What Is an Equation?
An equation is a mathematical statement that says two expressions are equal. It always contains an equals sign (=). The goal of solving an equation is to find the value of the unknown variable that makes the statement true.
Equations vs Expressions
| Expression | Equation | |
|---|---|---|
| Contains = | No | Yes |
| Example | 3x + 2 | 3x + 2 = 11 |
| Can solve for x? | No | Yes |
| Has a solution | No | Yes (one or more values) |
Types of Equations
| Type | Example | Highest Power |
|---|---|---|
| Linear | 2x + 3 = 9 | 1 |
| Quadratic | x squared - 5x + 6 = 0 | 2 |
| Cubic | x cubed = 8 | 3 |
| Simultaneous | x + y = 5 and 2x - y = 1 | Two unknowns |
The Balance Principle
Think of an equation as a balanced scale. To keep it balanced, any operation you perform on the left side must also be performed on the right side.
Subtract 5 from both sides: x + 5 - 5 = 12 - 5. Therefore x = 7.
Divide both sides by 3: 3y / 3 = 18 / 3. Therefore y = 6.
Checking a Solution
Substitute the answer back into the original equation. Both sides must give the same value.
Left side: 3 times 6 = 18. Right side: 18. Both sides equal. Solution verified!
Key Takeaways
- An equation has an equals sign; an expression does not.
- Treat an equation like a balanced scale — whatever you do to one side, do to the other.
- Always check your solution by substituting it back into the original equation.
Practice Questions
- Is 2x + 3 an equation or an expression? Explain why.
- Solve n + 9 = 20.
- Solve 4m = 28.
- Solve z - 7 = 15.
- Check whether x = 4 is a solution of 2x + 1 = 9.