How do determine if #x+y=-3, 2x+y=1# has no solution, one solution, or an Infinite number of solutions and find the solution?

Question

Julia Adams · Accepted Answer

#x = 4#, #y = 7#

arrange both equations so that #y# is on the left-hand side:

#x + y = -3#
#y = -3 - x#

#2x+y = 1#
#y = 1-2x#

#y = -3 - x = 1 - 2x#

#-3 - x = 1 - 2x#

#-3 = 1 - x#

#-4 = -x#

#-x = -4#

#x = 4#

#1 - (2*4) = 1-8 = -7#

#y = -7#

#x = 4, y = -7#

#(2*4) + -7 = 1#

#4 + -7 = -3#

this means that there is only one solution for each variable.
#x = 4# and #y = 7#.

the number of solutions could also be found by drawing the graphs of both:

the two graphs only meet once, and then extend in different directions.

therefore, it can be seen that the pair of simultaneous equations only has one solution.

the point of intersection between the two graphs is #(4,7)#.

#(4,7)# are the #xy#-coordinates, meaning that at this point, #x = 4# and #y = 7#.

this means that the solution is #x = 4# and #y = 7#.