Advanced Topics
In a nutshell: Systems of equations find where two equations are true at the same time.
## Solving Systems of Equations
Sometimes, you have to solve two equations at once! A **system of equations** is a set of two or more equations with the same variables. The solution is the point where both equations are true at the same time.
### Methods for Solving
- **Substitution**: Solve one equation for one variable, then substitute into the other.
- **Elimination**: Add or subtract equations to eliminate one variable.
- **Graphing**: Plot both equations on a graph and find their intersection.
### Real-Life Applications
Systems of equations help with things like planning budgets or finding when two things will be equal (like two trains meeting).
#### Tips
- Check solutions by plugging values into both equations.
- Organize your work to avoid mistakes.
y = mx + b
Examples
- If \( x + y = 10 \) and \( x - y = 2 \), add to get \( 2x = 12 \), so \( x = 6 \), then \( y = 4 \).
- Graph \( y = 2x \) and \( y = x + 3 \); they intersect at \( (3, 6) \).
Key terms
- System of Equations
- A set of equations with the same variables.
- Elimination
- A method for solving systems by adding or subtracting equations.