SOLUTION: {{{system( 2x + 3y = 4 , 3x + 4y = 5) }}} ( pretend the { is one big one )

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Question 308972: system%28+2x+%2B+3y+=+4+%2C+3x+%2B+4y+=+5%29+
( pretend the { is one big one )
Found 2 solutions by ichudov, jim_thompson5910:
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
+system%28+%0D%0A++++2%5Cx+%2B+3%5Cy+=+4%2C%0D%0A++++3%5Cx+%2B+4%5Cy+=+5+%29%0D%0A++We'll use substitution. After moving 3*y to the right, we get:
2%2Ax+=+4+-+3%2Ay, or x+=+4%2F2+-+3%2Ay%2F2. Substitute that
into another equation:
3%2A%284%2F2+-+3%2Ay%2F2%29+%2B+4%5Cy+=+5 and simplify: So, we know that y=2. Since x+=+4%2F2+-+3%2Ay%2F2, x=-1.

Answer: system%28+x=-1%2C+y=2+%29.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Start with the given system of equations:

system%282x%2B3y=4%2C3x%2B4y=5%29



Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

2x%2B3y=4 Start with the first equation


3y=4-2x Subtract 2x from both sides


3y=-2x%2B4 Rearrange the equation


y=%28-2x%2B4%29%2F%283%29 Divide both sides by 3


y=%28%28-2%29%2F%283%29%29x%2B%284%29%2F%283%29 Break up the fraction


y=%28-2%2F3%29x%2B4%2F3 Reduce



---------------------

Since y=%28-2%2F3%29x%2B4%2F3, we can now replace each y in the second equation with %28-2%2F3%29x%2B4%2F3 to solve for x



3x%2B4highlight%28%28%28-2%2F3%29x%2B4%2F3%29%29=5 Plug in y=%28-2%2F3%29x%2B4%2F3 into the second equation. In other words, replace each y with %28-2%2F3%29x%2B4%2F3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



3x%2B%284%29%28-2%2F3%29x%2B%284%29%284%2F3%29=5 Distribute 4 to %28-2%2F3%29x%2B4%2F3


3x-%288%2F3%29x%2B16%2F3=5 Multiply


%283%29%283x-%288%2F3%29x%2B16%2F3%29=%283%29%285%29 Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



9x-8x%2B16=15 Distribute and multiply the LCM to each side



x%2B16=15 Combine like terms on the left side


x=15-16Subtract 16 from both sides


x=-1 Combine like terms on the right side





-----------------First Answer------------------------------


So the first part of our answer is: x=-1









Since we know that x=-1 we can plug it into the equation y=%28-2%2F3%29x%2B4%2F3 (remember we previously solved for y in the first equation).



y=%28-2%2F3%29x%2B4%2F3 Start with the equation where y was previously isolated.


y=%28-2%2F3%29%28-1%29%2B4%2F3 Plug in x=-1


y=2%2F3%2B4%2F3 Multiply


y=2 Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is: y=2









-----------------Summary------------------------------

So our answers are:

x=-1 and y=2

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of 2x%2B3y=4 (red) and 3x%2B4y=5 (green) and the intersection of the lines (blue circle).