## Key Concepts

**How to determine whether an ordered pair is a solution to a system of linear equations**

- Substitute the ordered pair into each equation in the system.
- Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.

**Three possible outcomes for solutions to systems of equations**

**One Solution:**When a system of equations intersects at an ordered pair, the system has one solution.**Infinite Solutions:**Sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions.**No Solution:**When the lines that make up a system are parallel, there are no solutions because the two lines share no points in common.

**Based upon these outcomes, there are three types of systems of linear equations in two variables.**

- An
**independent system**has exactly*one solution*pair [latex]\left(x,y\right)[/latex]. The point where the two lines intersect is the only solution. - An
**inconsistent system**has*no solution*.The two lines are parallel and will never intersect. - A
**dependent system**has*infinitely many solutions*. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.

## Glossary

**System of linear equations** two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.

**The solution **to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

A **consistent system** of equations has at least one solution.

An **independent system** has a single solution.

A** dependent system** has an infinite number of solutions.

An **inconsistent** **system** is when there are no points common to both lines.