Question 123307


Start with the given system of equations:


{{{system(x-y=4,-5x+3y=6)}}}




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


{{{x-y=4}}} Start with the first equation



{{{-y=4-x}}}  Subtract {{{x}}} from both sides



{{{-y=-x+4}}} Rearrange the equation



{{{y=(-x+4)/(-1)}}} Divide both sides by {{{-1}}}



{{{y=((-1)/(-1))x+(4)/(-1)}}} Break up the fraction



{{{y=x-4}}} Reduce




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


Since {{{y=x-4}}}, we can now replace each {{{y}}} in the second equation with {{{x-4}}} to solve for {{{x}}}




{{{-5x+3highlight((x-4))=6}}} Plug in {{{y=x-4}}} into the first equation. In other words, replace each {{{y}}} with {{{x-4}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{-5x+(3)(1)x+(3)(-4)=6}}} Distribute {{{3}}} to {{{x-4}}}



{{{-5x+3x-12=6}}} Multiply



{{{-2x-12=6}}} Combine like terms on the left side



{{{-2x=6+12}}}Add 12 to both sides



{{{-2x=18}}} Combine like terms on the right side



{{{x=(18)/(-2)}}} Divide both sides by -2 to isolate x




{{{x=-9}}} Divide






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



So the first part of our answer is: {{{x=-9}}}










Since we know that {{{x=-9}}} we can plug it into the equation {{{y=x-4}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=x-4}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-9)-4}}} Plug in {{{x=-9}}}



{{{y=-9-4}}} Multiply



{{{y=-13}}} Combine like terms 




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



So the second part of our answer is: {{{y=-13}}}










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


So our answers are:


{{{x=-9}}} and {{{y=-13}}}


which form the point *[Tex \LARGE \left(-9,-13\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(-9,-13\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -15,10,-15,10,
  graph(500, 500, -15,10,-15,10, (4-1*x)/(-1), (6--5*x)/(3) ),
  blue(circle(-9,-13,0.1)),
  blue(circle(-9,-13,0.12)),
  blue(circle(-9,-13,0.15))
)
}}} graph of {{{x-y=4}}} (red) and {{{-5x+3y=6}}} (green)  and the intersection of the lines (blue circle).