Question 127283



Start with the given system of equations:


{{{system(3x+y=1,-6x+6y=-18)}}}




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


{{{3x+y=1}}} Start with the first equation



{{{y=1-3x}}}  Subtract {{{3x}}} from both sides



{{{y=-3x+1}}} Rearrange the equation



{{{y=(-3x+1)/(1)}}} Divide both sides by {{{1}}}



{{{y=((-3)/(1))x+(1)/(1)}}} Break up the fraction



{{{y=-3x+1}}} Reduce




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


Since {{{y=-3x+1}}}, we can now replace each {{{y}}} in the second equation with {{{-3x+1}}} to solve for {{{x}}}




{{{-6x+6highlight((-3x+1))=-18}}} Plug in {{{y=-3x+1}}} into the first equation. In other words, replace each {{{y}}} with {{{-3x+1}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{-6x+(6)(-3)x+(6)(1)=-18}}} Distribute {{{6}}} to {{{-3x+1}}}



{{{-6x-18x+6=-18}}} Multiply



{{{-24x+6=-18}}} Combine like terms on the left side



{{{-24x=-18-6}}}Subtract 6 from both sides



{{{-24x=-24}}} Combine like terms on the right side



{{{x=(-24)/(-24)}}} Divide both sides by -24 to isolate x




{{{x=1}}} Divide






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



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










Since we know that {{{x=1}}} we can plug it into the equation {{{y=-3x+1}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-3x+1}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-3(1)+1}}} Plug in {{{x=1}}}



{{{y=-3+1}}} Multiply



{{{y=-2}}} Combine like terms 




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



So the second part of our answer is: {{{y=-2}}}










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


So our answers are:


{{{x=1}}} and {{{y=-2}}}


which form the point *[Tex \LARGE \left(1,-2\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(1,-2\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (1-3*x)/(1), (-18--6*x)/(6) ),
  blue(circle(1,-2,0.1)),
  blue(circle(1,-2,0.12)),
  blue(circle(1,-2,0.15))
)
}}} graph of {{{3x+y=1}}} (red) and {{{-6x+6y=-18}}} (green)  and the intersection of the lines (blue circle).