Wetzel

Math 110

6.1(9.1) – Systems of Linear Equations in Two Variables

Systems of Linear Equations:

·        Two or more linear equations grouped together.

·        A solution to a system is a point that all the equations have in common or a solution to all of the equations in the system.

            Examples:  Determine if the given point is a solution to the system.

            1.)                                 2.) 

      ( -2, -7 )?                                             ( -1, 5 )?

·        One solution, No Solutions, or Infinitely Many Solutions.

·        Graphs…..

Graphing Method:   ( 1st method to solve a linear system )

                        To Solve:

1.      Graph each equation on the same graph.

2.      The solution is the point of intersection.

            Examples:  Solve using the Graphing Method.

            1.)                                          2.) 

            Advantages vs. Disadvantages of the Graphing Method.

Substitution Method:  ( 2nd method to solve a linear system )

            To Solve:  ( 2 equations with 2 variables )

                        1.)  Solve for one of the variables in one of the equations.

                        2.)  Substitute this value into the other equation and solve for the remaining variable.

                        3.)  Substitute this value into one of the original equations and solve for the remaining variable.

                        4.)  Write the answer as an ordered pair.

            Examples:

            1.)                                             2.) 

·        Infinitely Many Solutions:   All variables cancel and the remaining equation is TRUE.

·        No Solution:   All variables cancel and the remaining equation is FALSE.

Elimination Method:  ( 3rd method to solve a linear system )

            To Solve:  ( 2 equations with 2 variables )

                        1.)  Write all equations in standard form.   ax + by = c

                        2.)  Get opposite coefficients for the x or y variables by multiplying one or both of the equations by a constant.

                        3.)  Add the two equations together  (The variable with the opposite coefficients should cancel) and SOLVE for the remaining variable.

                        4.)  Substitute this value into one of the original equations and solve for the other variable.

                        5.)  Write the answer as an ordered pair.

            Examples:

            1.)                           2.) 

            3.)                                     4.) 

·        Infinitely Many Solutions:   All variables cancel and the remaining equation is TRUE.

·        No Solution:   All variables cancel and the remaining equation is FALSE.

HW:     6^th Ed. & 5^th Ed.  p.524  #3-33odd