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Solving Systems of Equations



It often makes sense to consider two or more equations at the same time. This collection of equations is called a system. A solution for one of the equations is not necessarily a solution for any of the other equations in a system. But when they all have a solution in common, that is a solution for the system. Not all systems have solutions. Linear systems can have zero, one, or an infinite number of solutions. The possible number of solutions in a nonlinear system depends on the types of equations in the system. Algebraic methods or graphing methods can be used to solve systems of equations.

At A Glance

  • A system of linear equations can have zero, one, or infinitely many solutions. For two variables, they are represented graphically as parallel lines, intersecting lines, or lines that coincide. For three variables, they are represented as intersections of planes.
  • The solution of a system of linear equations can be estimated from a graph and then confirmed by substituting the values into the equations.
  • A system of linear equations can be solved by solving an equation for one variable and substituting the resulting expression in the other equation(s).
  • Equations in a system can be added, subtracted, and/or multiplied by a nonzero constant in order to eliminate a variable.
  • Graphs of nonlinear systems of equations can be used to determine the number of solutions and estimate their values.
  • Nonlinear systems can be solved by algebraic methods, such as substitution and elimination.