Summary: Solutions to Systems of Equations

Key Concepts

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

  1. Substitute the ordered pair into each equation in the system.
  2. 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.


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.