How do you determine the solution in terms of a System of Linear Equations for -2x + 3y =5 and ax - y = 1?

1 Answer
Oct 7, 2015

Answer:

This system has:

  • No solution if #a=2/3#
  • One solution : #{(x=(-8)/(2-3a)), (y=(-2-5a)/(2-3a)):}# if #a!=2/3#

Explanation:

To find the connection between value of a parameter #a# and the number of solutions of a linear system you can use the Cramer's Rule.

It can be written as follows:

Let there be a system of 2 linear equations:

#{(a_1x+b_1y=c_1),(a_2x+b_2y=c_2):}#

Let

#W=a_1*b_2-b_1*a_2#

#W_x=c_1b_2-b_1c_2#

#W_y=a_1c_2-c_1a_2#

Then the system has:

  • One solution #{(x=W_x/W),(y=W_y/W):} iff W!=0#
  • No solutions #iff W=0# and #W_x!=0# or #W_y!=0#
  • Infinitely many solutions #iff W=0# and #W_x=0# and #W_y=0#

This rule can be expanded for any system of #n# equations with #n# variables, then you have #W# and for every variable #x_i# you have a corresponding determinant #W_(x_i)#, and the rule says that:

  1. If #W!=0# system has exactly one solution: #x_i=(W_{x_i})/W# for #1<=i<=n#

  2. If #W=0# and any of #W_{x_i}# is not zero, then system has no solutions

  3. If #W=0# and all #W_{x_i}# are zeros, then the system has infinitely many solutions.