SOLUTION: Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10

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Question 124907: Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10
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%283x-4y=8%2C6x-2y=10%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-4y=8 Start with the first equation


-4y=8-3x Subtract 3x from both sides


-4y=-3x%2B8 Rearrange the equation


y=%28-3x%2B8%29%2F%28-4%29 Divide both sides by -4


y=%28%28-3%29%2F%28-4%29%29x%2B%288%29%2F%28-4%29 Break up the fraction


y=%283%2F4%29x-2 Reduce



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

Since y=%283%2F4%29x-2, we can now replace each y in the second equation with %283%2F4%29x-2 to solve for x



6x-2highlight%28%28%283%2F4%29x-2%29%29=10 Plug in y=%283%2F4%29x-2 into the first equation. In other words, replace each y with %283%2F4%29x-2. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



6x%2B%28-2%29%283%2F4%29x%2B%28-2%29%28-2%29=10 Distribute -2 to %283%2F4%29x-2


6x-%286%2F4%29x%2B4=10 Multiply


%284%29%286x-%286%2F4%29x%2B4%29=%284%29%2810%29 Multiply both sides by the LCM of 4. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



24x-6x%2B16=40 Distribute and multiply the LCM to each side



18x%2B16=40 Combine like terms on the left side


18x=40-16Subtract 16 from both sides


18x=24 Combine like terms on the right side


x=%2824%29%2F%2818%29 Divide both sides by 18 to isolate x



x=4%2F3 Reduce





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


So the first part of our answer is: x=4%2F3









Since we know that x=4%2F3 we can plug it into the equation y=%283%2F4%29x-2 (remember we previously solved for y in the first equation).



y=%283%2F4%29x-2 Start with the equation where y was previously isolated.


y=%283%2F4%29%284%2F3%29-2 Plug in x=4%2F3


y=12%2F12-2 Multiply


y=-1 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=-1









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

So our answers are:

x=4%2F3 and y=-1

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-4y=8 (red) and 6x-2y=10 (green) and the intersection of the lines (blue circle).