Question 550539



Start with the given system of equations:


{{{system(3x+9y=57,-7x+y=65)}}}



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 second equation


{{{-7x+y=65}}} Start with the second equation



{{{y=65+7x}}} Add {{{7x}}} to both sides



{{{y=+7x+65}}} Rearrange the equation



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


Since {{{y=7x+65}}}, we can now replace each {{{y}}} in the first equation with {{{7x+65}}} to solve for {{{x}}}




{{{3x+9highlight((7x+65))=57}}} Plug in {{{y=7x+65}}} into the second equation. In other words, replace each {{{y}}} with {{{7x+65}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{3x+(9)(7)x+(9)(65)=57}}} Distribute {{{9}}} to {{{7x+65}}}



{{{3x+63x+585=57}}} Multiply



{{{66x+585=57}}} Combine like terms on the left side



{{{66x=57-585}}}Subtract 585 from both sides



{{{66x=-528}}} Combine like terms on the right side



{{{x=(-528)/(66)}}} Divide both sides by 66 to isolate x




{{{x=-8}}} Divide






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



So the first part of our answer is: {{{x=-8}}}










Since we know that {{{x=-8}}} we can plug it into the equation {{{y=7x+65}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=7x+65}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=7(-8)+65}}} Plug in {{{x=-8}}}



{{{y=-56+65}}} Multiply



{{{y=9}}} Combine like terms 




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



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










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


So our answers are:


{{{x=-8}}} and {{{y=9}}}


which form the point *[Tex \LARGE \left(-8,9\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(-8,9\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (65--7*x)/(1), (57-3*x)/(9) ),
  blue(circle(-8,9,0.1)),
  blue(circle(-8,9,0.12)),
  blue(circle(-8,9,0.15))
)
}}} graph of {{{-7x+y=65}}} (red) and {{{3x+9y=57}}} (green)  and the intersection of the lines (blue circle).