Question

System of linear equations

vocabulary term definition

Answer 1

A system of linear equations is a set of two or more linear equations that share the same variables. The solutions to a system of linear equations are the values for the variables that satisfy all equations in the system simultaneously.

### Key Terms:
- **Linear Equation**: An equation that describes a straight line when graphed, typically in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
- **Solution**: A set of values assigned to the variables that makes all equations in the system true.
- **Consistent System**: A system of linear equations that has at least one solution.
- **Inconsistent System**: A system of linear equations that has no solutions.
- **Dependent System**: A system where there are infinitely many solutions, often because the equations represent the same line.
- **Independent System**: A system with exactly one solution, where the equations represent different lines that intersect at a single point.

### Methods of Solving:
1. **Graphical Method**: Plotting each equation on a graph to find the intersection point.
2. **Substitution Method**: Solving one equation for a variable and substituting that expression into the other equations.
3. **Elimination Method**: Adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variables.
4. **Matrix Method**: Using matrices and determinants (typically through techniques like Gaussian elimination) to solve the system.

Systems of linear equations have applications in various fields, including economics, engineering, physics, and statistics, wherever relationships between variables need to be analyzed.