SOLUTION: solve the following simultaneous equations 3x+2y=12 5x-2y+4 3x+5y=26 2x+3y=16

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Question 134322: solve the following simultaneous equations
3x+2y=12
5x-2y+4
3x+5y=26
2x+3y=16
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that the first two are the first system of equations. Do you want to use substitution?

# 1




Start with the given system of equations:

system%283x%2B2y=12%2C5x-2y=4%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

3x%2B2y=12 Start with the first equation


2y=12-3x Subtract 3x from both sides


2y=-3x%2B12 Rearrange the equation


y=%28-3x%2B12%29%2F%282%29 Divide both sides by 2


y=%28%28-3%29%2F%282%29%29x%2B%2812%29%2F%282%29 Break up the fraction


y=%28-3%2F2%29x%2B6 Reduce



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

Since y=%28-3%2F2%29x%2B6, we can now replace each y in the second equation with %28-3%2F2%29x%2B6 to solve for x



5x-2highlight%28%28%28-3%2F2%29x%2B6%29%29=4 Plug in y=%28-3%2F2%29x%2B6 into the first equation. In other words, replace each y with %28-3%2F2%29x%2B6. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



5x%2B%28-2%29%28-3%2F2%29x%2B%28-2%29%286%29=4 Distribute -2 to %28-3%2F2%29x%2B6


5x%2B%286%2F2%29x-12=4 Multiply


%282%29%285x%2B%286%2F2%29x-12%29=%282%29%284%29 Multiply both sides by the LCM of 2. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



10x%2B6x-24=8 Distribute and multiply the LCM to each side



16x-24=8 Combine like terms on the left side


16x=8%2B24Add 24 to both sides


16x=32 Combine like terms on the right side


x=%2832%29%2F%2816%29 Divide both sides by 16 to isolate x



x=2 Divide





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


So the first part of our answer is: x=2









Since we know that x=2 we can plug it into the equation y=%28-3%2F2%29x%2B6 (remember we previously solved for y in the first equation).



y=%28-3%2F2%29x%2B6 Start with the equation where y was previously isolated.


y=%28-3%2F2%29%282%29%2B6 Plug in x=2


y=-6%2F2%2B6 Multiply


y=3 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=3









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

So our answers are:

x=2 and y=3

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 3x%2B2y=12 (red) and 5x-2y=4 (green) and the intersection of the lines (blue circle).





# 2





Start with the given system of equations:

system%283x%2B5y=26%2C2x%2B3y=16%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

3x%2B5y=26 Start with the first equation


5y=26-3x Subtract 3x from both sides


5y=-3x%2B26 Rearrange the equation


y=%28-3x%2B26%29%2F%285%29 Divide both sides by 5


y=%28%28-3%29%2F%285%29%29x%2B%2826%29%2F%285%29 Break up the fraction


y=%28-3%2F5%29x%2B26%2F5 Reduce



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

Since y=%28-3%2F5%29x%2B26%2F5, we can now replace each y in the second equation with %28-3%2F5%29x%2B26%2F5 to solve for x



2x%2B3highlight%28%28%28-3%2F5%29x%2B26%2F5%29%29=16 Plug in y=%28-3%2F5%29x%2B26%2F5 into the first equation. In other words, replace each y with %28-3%2F5%29x%2B26%2F5. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



2x%2B%283%29%28-3%2F5%29x%2B%283%29%2826%2F5%29=16 Distribute 3 to %28-3%2F5%29x%2B26%2F5


2x-%289%2F5%29x%2B78%2F5=16 Multiply


%285%29%282x-%289%2F5%29x%2B78%2F5%29=%285%29%2816%29 Multiply both sides by the LCM of 5. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



10x-9x%2B78=80 Distribute and multiply the LCM to each side



x%2B78=80 Combine like terms on the left side


x=80-78Subtract 78 from both sides


x=2 Combine like terms on the right side





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


So the first part of our answer is: x=2









Since we know that x=2 we can plug it into the equation y=%28-3%2F5%29x%2B26%2F5 (remember we previously solved for y in the first equation).



y=%28-3%2F5%29x%2B26%2F5 Start with the equation where y was previously isolated.


y=%28-3%2F5%29%282%29%2B26%2F5 Plug in x=2


y=-6%2F5%2B26%2F5 Multiply


y=4 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=4









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

So our answers are:

x=2 and y=4

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 3x%2B5y=26 (red) and 2x%2B3y=16 (green) and the intersection of the lines (blue circle).