How do I work this problem?

Question

Consider the following system of equations:

2x − 3y = 2,
4x + 2y = −1
−5x − 6y = d
For what value of d does the above system have a solution? For this value of d, graph all three equations together.
What do you notice?

Answer 1

For a fixed value of d,  this set is over determined.   There are three equations but only two unknowns.  

One approach is to solve the first two equations for x and y.   This can be done by standard methods.   The result is:

 

     x = 1/16     y = -10/16

 

Plugging in these values of x and y into the third equation results in d = 55/16.

 

This is the only value of d for which the solution set for the three equations is not the empty set (that is no solution)

 

For d = 55/16     the graphs of the three lines all meet at  x = 1/16,  y = -10/16