This problem include Question 1 from Quiz 2 for Sections 1 and 2.


Problem. Solve for x, y

ax + y = b
x+ay = c

where a,b,c are given numbers.

Solution. 
Solve the first equation for y:

y= b - ax.
Use it to eliminate y from the second equation.

x+a(b-ax) = c
or (in standard form)
(1-a2)x = c - ab.
Solve this for x:

x=   (c-ab)/(1-a2) when a =/  1, -1,
no solutions when a=1 and c =/ b or   a = -1  and  c =/ -b,
x is arbitrary otherwise.
Substitute this to  y= b - ax.:
x=   (c-ab)/(1-a2) , y = b - a (c-ab)/(1-a2)    when a =/  1, -1,
no solutions when a=1 and c =/ b or   a = -1  and  c =/ -b,
y= b - ax   and  x is arbitrary otherwise.