Question 574295


Start with the given system of equations:


{{{system(-7x+y=47,7x+9y=3)}}}




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+y=47}}} Start with the first equation



{{{y=47+7x}}} Add {{{7x}}} to both sides



{{{y=+7x+47}}} Rearrange the equation


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


Since {{{y=7x+47}}}, we can now replace each {{{y}}} in the second equation with {{{7x+47}}} to solve for {{{x}}}




{{{7x+9highlight((7x+47))=3}}} Plug in {{{y=7x+47}}} into the second equation. In other words, replace each {{{y}}} with {{{7x+47}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{7x+(9)(7)x+(9)(47)=3}}} Distribute {{{9}}} to {{{7x+47}}}



{{{7x+63x+423=3}}} Multiply



{{{70x+423=3}}} Combine like terms on the left side



{{{70x=3-423}}}Subtract 423 from both sides



{{{70x=-420}}} Combine like terms on the right side



{{{x=(-420)/(70)}}} Divide both sides by 70 to isolate x




{{{x=-6}}} Divide






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



So the first part of our answer is: {{{x=-6}}}










Since we know that {{{x=-6}}} we can plug it into the equation {{{y=7x+47}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=7x+47}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=7(-6)+47}}} Plug in {{{x=-6}}}



{{{y=-42+47}}} Multiply



{{{y=5}}} Combine like terms 




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



So the second part of our answer is: {{{y=5}}}










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


So our answers are:


{{{x=-6}}} and {{{y=5}}}


which form the point *[Tex \LARGE \left(-6,5\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(-6,5\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  grid(1),
  graph(500, 500, -10,10,-10,10, (47--7*x)/(1), (3-7*x)/(9) ),
  blue(circle(-6,5,0.1)),
  blue(circle(-6,5,0.12)),
  blue(circle(-6,5,0.15))
)
}}} graph of {{{-7x+y=47}}} (red) and {{{7x+9y=3}}} (green)  and the intersection of the lines (blue circle).