Linear System Equation with Two Variables

 

 

Motivation: 

                        We start with the following problem.  If the sum of two numbers is 150 and the difference is 50, what are the two numbers?  There are many ways to find out the solution. However, it is straightforward to convert the daily language into mathematical equation and focus on solving the equation.  So,  let’s denote the two numbers as x and y. And we have

                                Equation I:       x + y = 150

                                Equation II:      x – y = 50

 

                       Our goal turns out to be finding x and y such that the two equations can be satisfied simultaneously.

 

                       If we add up equation I and equation II:

 

                            Equation I + Equation II:

                                                            (x+y)+(x-y) = 150 + 50

                                                             2x = 200

                                                               x = 100

                                   With this result,  we will find out  y=50 .   Thus, the two numbers are 100 and 50.

 

                     In order to solve this type of problems more efficiently,  we would like to develop some skills on the system equation below:

 

                                    

 

                    Once we figure out all the scenarios of this system equation,  the solutions to similar questions can be easily solved.

 

 

Solution to the Linear System Equation with Two Variables

 

                                   

 

                      We multiply the first equation by  d and the second equation by b:

                                   Eq 3:       d(ax+by) = de              

                                   Eq 4:       b(cx+dy) = bf

 

                        Eq 3 – Eq 4:

                                                adx – bcx = de -bf

                                                (ad-bc)x = (de-bf)

 

                          It is similar to what we have for the equation with one variable before:

 

                        Case I:    ad-cb   0    

 

                                           In this case, we can divide both sides by (ad-bc). Then

                                               x =

 

                                          From that, we know

                                                y=

 

                         Case II:  (ad-cb) = 0 and (de-fb)=0

 

                               In this case, there are infinitely many solutions to this system equation.

 

                                        Please notice that  if c, d, f are not 0, we can use the following to

                                        check if the system equation is with infinitely many solutions:

                                               

 

                         Case III:  (ad-cb) = 0 but (de-fb)  0

 

                                           In this case, there is no solution to this system equation.

                                          Please notice if c, d, f are not 0, we can use the following to

                                          check if there is no solution to the system equation:

                                           

 

 

Example 1:  

                

                    This system equation has infinitely many solutions.

                    As a matter of fact, you find out that the two equations are actually one equation.

 

 

 

 

 

 

 

Example 2:

                

                    This system equation has no solution.

                    Actually, the second one can be simplified as

                            3x+2y = 5

                   Along with the first one 3x+2y=6,  we know that it is impossible to satisfy the two equations at the same time.

 

 

Example 3:

                  

               The solution is: x=-14, y=24.