Question 131028
I'll do the first two to get you started




# 1





Start with the given system of equations:


{{{system(7x+3y=-28,-2x+y=21)}}}




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


{{{7x+3y=-28}}} Start with the first equation



{{{3y=-28-7x}}}  Subtract {{{7x}}} from both sides



{{{3y=-7x-28}}} Rearrange the equation



{{{y=(-7x-28)/(3)}}} Divide both sides by {{{3}}}



{{{y=((-7)/(3))x+(-28)/(3)}}} Break up the fraction



{{{y=(-7/3)x-28/3}}} Reduce




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


Since {{{y=(-7/3)x-28/3}}}, we can now replace each {{{y}}} in the second equation with {{{(-7/3)x-28/3}}} to solve for {{{x}}}




{{{-2x+highlight(((-7/3)x-28/3))=21}}} Plug in {{{y=(-7/3)x-28/3}}} into the first equation. In other words, replace each {{{y}}} with {{{(-7/3)x-28/3}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{(3)(-2x-(7/3)x-28/3)=(3)(21)}}} 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>)




{{{-6x-7x-28=63}}} Distribute and multiply the LCM to each side




{{{-13x-28=63}}} Combine like terms on the left side



{{{-13x=63+28}}}Add 28 to both sides



{{{-13x=91}}} Combine like terms on the right side



{{{x=(91)/(-13)}}} Divide both sides by -13 to isolate x




{{{x=-7}}} Divide






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



So the first part of our answer is: {{{x=-7}}}










Since we know that {{{x=-7}}} we can plug it into the equation {{{y=(-7/3)x-28/3}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-7/3)x-28/3}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-7/3)(-7)-28/3}}} Plug in {{{x=-7}}}



{{{y=49/3-28/3}}} Multiply



{{{y=7}}} 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=7}}}










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


So our answers are:


{{{x=-7}}} and {{{y=7}}}


which form the point *[Tex \LARGE \left(-7,7\right)] 









Now let's graph the two equations (if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)



From the graph, we can see that the two equations intersect at *[Tex \LARGE \left(-7,7\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-28-7*x)/(3), (21--2*x)/(1) ),
  blue(circle(-7,7,0.1)),
  blue(circle(-7,7,0.12)),
  blue(circle(-7,7,0.15))
)
}}} graph of {{{7x+3y=-28}}} (red) and {{{-2x+y=21}}} (green)  and the intersection of the lines (blue circle).




<hr>







Start with the given system of equations:


{{{system(2m+n=-7,m-8m=73)}}}




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





So let's isolate n in the first equation


{{{2m+n=-7}}} Start with the first equation



{{{n=-7-2m}}}  Subtract {{{2m}}} from both sides



{{{n=-2m-7}}} Rearrange the equation



{{{n=(-2m-7)/(1)}}} Divide both sides by {{{1}}}



{{{n=((-2)/(1))m+(-7)/(1)}}} Break up the fraction



{{{n=-2m-7}}} Reduce




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


Since {{{n=-2m-7}}}, we can now replace each {{{n}}} in the second equation with {{{-2m-7}}} to solve for {{{m}}}




{{{m-8m=73}}} Plug in {{{n=-2m-7}}} into the first equation. In other words, replace each {{{n}}} with {{{-2m-7}}}. Notice we've eliminated the {{{n}}} variables. So we now have a simple equation with one unknown.




{{{m+(-8)(-2)m+(-8)(-7)=73}}} Distribute {{{-8}}} to {{{-2m-7}}}



{{{m+16m+56=73}}} Multiply



{{{17m+56=73}}} Combine like terms on the left side



{{{17m=73-56}}}Subtract 56 from both sides



{{{17m=17}}} Combine like terms on the right side



{{{m=(17)/(17)}}} Divide both sides by 17 to isolate m




{{{m=1}}} Divide






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



So the first part of our answer is: {{{m=1}}}










Since we know that {{{m=1}}} we can plug it into the equation {{{n=-2m-7}}} (remember we previously solved for {{{n}}} in the first equation).




{{{n=-2m-7}}} Start with the equation where {{{n}}} was previously isolated.



{{{n=-2(1)-7}}} Plug in {{{m=1}}}



{{{n=-2-7}}} Multiply



{{{n=-9}}} Combine like terms 




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



So the second part of our answer is: {{{n=-9}}}










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


So our answers are:


{{{m=1}}} and {{{n=-9}}}


which form the point *[Tex \LARGE \left(1,-9\right)] (note: simply replace m with x and replace n with y)









Now let's graph the two equations (if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)



From the graph, we can see that the two equations intersect at *[Tex \LARGE \left(1,-9\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-7-2*x)/(1), (73-1*x)/(-8) ),
  blue(circle(1,-9,0.1)),
  blue(circle(1,-9,0.12)),
  blue(circle(1,-9,0.15))
)
}}} graph of {{{2m+n=-7}}} (red) and {{{m-8m=73}}} (green)  and the intersection of the lines (blue circle).