Expert: Paul Klarreich - 12/9/2007

Question
Can you show me how to solve the system of linear equations by using the substitution method? I also need to know if it is independent, dependent or inconsistent??? The equations are:
4x - 3y = 4
12x - 9y = 12  

Answer
Questioner:   Amina
Category:  Advanced Math
Private:  No
 
Subject:  Intermediate Algebra: Solving the system of linear equations
Question:  Can you show me how to solve the system of linear equations by using the substitution method? I also need to know if it is independent, dependent or inconsistent??? The equations are:


4x - 3y = 4
12x - 9y = 12
....................................
Hi, Amina,

Solve by substitution, you say?  That wouldn't be my choice, but......

4x - 3y = 4    (A)
12x - 9y = 12  (B)

Solve one equation, say A, for one of the variables:

4x = 4 + 3y
    4 + 3y
x = -------
      4

Substitute into the other.
    4 + 3y
12(--------) - 9y = 12
      4

3(4 + 3y) - 9y = 12

12 + 9y - 9y = 12

12 = 12

This is an IDENTITY; true for all y; in this case the system is DEPENDENT; i.e. the two graphs are actually the same graph.

Note: change the '12' in (B) to some other number, like 17, perhaps, and the last lines would say:

12 = 17

This is a CONTRADICTION; false for all y; in this case the system is INCONSISTENT; i.e. the two graphs are parallel lines and have no common solution.


Most of the time you get something like:

3y = 12

This is a CONDITIONAL -- an equation that is true for some but not all y, in this case y = 4; now the two graphs meet at one and only one point, the system is CONSISTENT and everyone is happy.