Question 153187
I'll do the first two to get you started:



1)



Start with the given system of equations:


{{{system(3x-2y=8,-12x+8y=32)}}}



Let's use substitution to solve the system



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-2y=8}}} Start with the first equation



{{{-2y=8-3x}}}  Subtract {{{3x}}} from both sides



{{{-2y=-3x+8}}} Rearrange the equation



{{{y=(-3x+8)/(-2)}}} Divide both sides by {{{-2}}}



{{{y=((-3)/(-2))x+(8)/(-2)}}} Break up the fraction



{{{y=(3/2)x-4}}} Reduce




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Since {{{y=(3/2)x-4}}}, we can now replace each {{{y}}} in the second equation with {{{(3/2)x-4}}} to solve for {{{x}}}




{{{-12x+8highlight(((3/2)x-4))=32}}} Plug in {{{y=(3/2)x-4}}} into the first equation. In other words, replace each {{{y}}} with {{{(3/2)x-4}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{-12x+(8)(3/2)x+(8)(-4)=32}}} Distribute {{{8}}} to {{{(3/2)x-4}}}



{{{-12x+(24/2)x-32=32}}} Multiply



{{{(2)(-12x+(24/2)x-32)=(2)(32)}}} 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 <a href=http://www.algebra.com/algebra/homework/divisibility/least-common-multiple.solver>solver</a>)




{{{-24x+24x-64=64}}} Distribute and multiply the LCM to each side




{{{-64=64}}} Combine like terms on the left side



{{{0=64+64}}}Add 64 to both sides



{{{0=128}}} Combine like terms on the right side



{{{0=128}}} Simplify


Since this equation is <font size=4><b>never</b></font> true for any x value, this means there are no solutions.



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2)






Start with the given system of equations:

{{{system(7x-5y=14,-4x+y=27)}}}




Let's use elimination to solve the system



{{{5(-4x+y)=5(27)}}} Multiply the both sides of the second equation by 5.



{{{-20x+5y=135}}} Distribute and multiply.



So we have the new system of equations:

{{{system(7x-5y=14,-20x+5y=135)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(7x-5y)+(-20x+5y)=(14)+(135)}}}



{{{(7x+-20x)+(-5y+5y)=14+135}}} Group like terms.



{{{-13x+0y=149}}} Combine like terms. Notice how the y terms cancel out.



{{{-13x=149}}} Simplify.



{{{x=(149)/(-13)}}} Divide both sides by {{{-13}}} to isolate {{{x}}}.



{{{x=-149/13}}} Reduce.



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{{{7x-5y=14}}} Now go back to the first equation.



{{{7(-149/13)-5y=14}}} Plug in {{{x=-149/13}}}.



{{{-1043/13-5y=14}}} Multiply.



{{{-5y=14+1043/13}}} Add {{{1043/13}}} to both sides.



{{{-5y=1225/13}}} Combine like terms



{{{y=-245/13}}} Divide both sides by -5.





So our answer is {{{x=-149/13}}} and {{{y=-245/13}}}.



Which form the ordered pair *[Tex \LARGE \left(-\frac{149}{13},-\frac{245}{13}\right)].