Question 146605


Start with the given system of equations:


{{{system(9x+7y=-47,-5x+y=31)}}}




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


{{{-5x+y=31}}} Start with the second equation



{{{y=31+5x}}} Add {{{5x}}} to both sides



{{{y=5x+31}}} Rearrange the equation






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


Since {{{y=5x+31}}}, we can now replace each {{{y}}} in the first equation with {{{5x+31}}} to solve for {{{x}}}




{{{9x+7(5x+31)=-47}}} Plug in {{{y=5x+31}}} into the first equation. In other words, replace each {{{y}}} with {{{5x+31}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{9x+(7)(5)x+(7)(31)=-47}}} Distribute {{{7}}} to {{{5x+31}}}



{{{9x+35x+217=-47}}} Multiply



{{{44x+217=-47}}} Combine like terms on the left side



{{{44x=-47-217}}}Subtract 217 from both sides



{{{44x=-264}}} Combine like terms on the right side



{{{x=(-264)/(44)}}} Divide both sides by 44 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=5x+31}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=5x+31}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=5(-6)+31}}} Plug in {{{x=-6}}}



{{{y=-30+31}}} Multiply



{{{y=1}}} Combine like terms 




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



So the second part of our answer is: {{{y=1}}}










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


So our answers are:


{{{x=-6}}} and {{{y=1}}}


which form the point *[Tex \LARGE \left(-6,1\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(-6,1\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-47-9*x)/(7),(31--5*x)/(1) ),
  blue(circle(-6,1,0.1)),
  blue(circle(-6,1,0.12)),
  blue(circle(-6,1,0.15))
)
}}} graph of {{{9x+7y=-47}}} (red) and {{{-5x+y=31}}} (green)  and the intersection of the lines (blue circle).