How do you solve the following linear system: # 2x + 4y = 7 , 3x + y = 4 #?

1 Answer
Ken C.
Mar 19, 2016

#(9/10, 13/10)#
Refer below for explanation.

Explanation:

First, arrange them like this so you can compare terms:
#2x+4y=7#
#3x+y=4#

When we solve linear systems like this, we always look for ways to cancel one variable. We can see that if we multiply #3x+y=4# by #-4#, we will get a #-4y# term. And then, if we add the new equation (with #-4y# in it) to the other equation, the #4y# and #-4y# will cancel. Watch:
#-4(3x+y=4)->-12x-4y=-16#

#-12xcancel(-4y)=-16#
#+2xcancel(+4y)=7#
#-----#
#-10x=-9#
#x=9/10#

We can now use this #x# value to solve for #y#, like so:
#2x+4y=7#
#2(9/10)+4y=7#
#9/5+4y=7#
#4y=7-9/5#
#4y=26/5#
#y=13/10#

Therefore our solution is #(9/10, 13/10)#. We can confirm this result in several ways. One, we can substitute these values into the original equations:
#2x+4y=7->2(9/10)+4(13/10)=7->7=7#
#3x+y=4->3(9/10)+(13/10)=4->4=4#

We can also look at the graph of these two equations, and confirm that they intersect at the point #(9/10, 13/10)#.
enter image source here
Note that #9/10=0.9# and #13/10=1.3#.

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