Question 124486

Do you want to solve by substitution?



Start with the given system of equations:


{{{system(1000x+30y=500,x-2y=11)}}}




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


{{{1000x+30y=500}}} Start with the first equation



{{{30y=500-1000x}}}  Subtract {{{1000x}}} from both sides



{{{30y=-1000x+500}}} Rearrange the equation



{{{y=(-1000x+500)/(30)}}} Divide both sides by {{{30}}}



{{{y=((-1000)/(30))x+(500)/(30)}}} Break up the fraction



{{{y=(-100/3)x+50/3}}} Reduce




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Since {{{y=(-100/3)x+50/3}}}, we can now replace each {{{y}}} in the second equation with {{{(-100/3)x+50/3}}} to solve for {{{x}}}




{{{x-2highlight(((-100/3)x+50/3))=11}}} Plug in {{{y=(-100/3)x+50/3}}} into the first equation. In other words, replace each {{{y}}} with {{{(-100/3)x+50/3}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{x+(-2)(-100/3)x+(-2)(50/3)=11}}} Distribute {{{-2}}} to {{{(-100/3)x+50/3}}}



{{{x+(200/3)x-100/3=11}}} Multiply



{{{(3)(1x+(200/3)x-100/3)=(3)(11)}}} 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 <a href=http://www.algebra.com/algebra/homework/divisibility/least-common-multiple.solver>solver</a>)




{{{3x+200x-100=33}}} Distribute and multiply the LCM to each side




{{{203x-100=33}}} Combine like terms on the left side



{{{203x=33+100}}}Add 100 to both sides



{{{203x=133}}} Combine like terms on the right side



{{{x=(133)/(203)}}} Divide both sides by 203 to isolate x




{{{x=19/29}}} Reduce






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



So the first part of our answer is: {{{x=19/29}}}










Since we know that {{{x=19/29}}} we can plug it into the equation {{{y=(-100/3)x+50/3}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-100/3)x+50/3}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-100/3)(19/29)+50/3}}} Plug in {{{x=19/29}}}



{{{y=-1900/87+50/3}}} Multiply



{{{y=-150/29}}} Combine like terms and reduce.  (note: if you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>)




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



So the second part of our answer is: {{{y=-150/29}}}










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


So our answers are:


{{{x=19/29}}} and {{{y=-150/29}}}


which form the point *[Tex \LARGE \left(\frac{19}{29},-\frac{150}{29}\right)]