Question 127286



Start with the given system of equations:


{{{system(-5x+2y=-10,3x-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


{{{-5x+2y=-10}}} Start with the first equation



{{{2y=-10+5x}}} Add {{{5x}}} to both sides



{{{2y=+5x-10}}} Rearrange the equation



{{{y=(+5x-10)/(2)}}} Divide both sides by {{{2}}}



{{{y=((+5)/(2))x+(-10)/(2)}}} Break up the fraction



{{{y=(5/2)x-5}}} Reduce




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


Since {{{y=(5/2)x-5}}}, we can now replace each {{{y}}} in the second equation with {{{(5/2)x-5}}} to solve for {{{x}}}




{{{3x-6highlight(((5/2)x-5))=-18}}} Plug in {{{y=(5/2)x-5}}} into the first equation. In other words, replace each {{{y}}} with {{{(5/2)x-5}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{3x+(-6)(5/2)x+(-6)(-5)=-18}}} Distribute {{{-6}}} to {{{(5/2)x-5}}}



{{{3x-(30/2)x+30=-18}}} Multiply



{{{(2)(3x-(30/2)x+30)=(2)(-18)}}} Multiply both sides by the LCM of 2. This will eliminate the fractions  (note: if you need help with finding the LCM, check out this <a href=http://www.algebra.com/algebra/homework/divisibility/least-common-multiple.solver>solver</a>)




{{{6x-30x+60=-36}}} Distribute and multiply the LCM to each side




{{{-24x+60=-36}}} Combine like terms on the left side



{{{-24x=-36-60}}}Subtract 60 from both sides



{{{-24x=-96}}} Combine like terms on the right side



{{{x=(-96)/(-24)}}} Divide both sides by -24 to isolate x




{{{x=4}}} Divide






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



So the first part of our answer is: {{{x=4}}}










Since we know that {{{x=4}}} we can plug it into the equation {{{y=(5/2)x-5}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(5/2)x-5}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(5/2)(4)-5}}} Plug in {{{x=4}}}



{{{y=20/2-5}}} Multiply



{{{y=5}}} Combine like terms and reduce.  (note: if you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>)




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



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










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


So our answers are:


{{{x=4}}} and {{{y=5}}}


which form the point *[Tex \LARGE \left(4,5\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(4,5\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-10--5*x)/(2), (-18-3*x)/(-6) ),
  blue(circle(4,5,0.1)),
  blue(circle(4,5,0.12)),
  blue(circle(4,5,0.15))
)
}}} graph of {{{-5x+2y=-10}}} (red) and {{{3x-6y=-18}}} (green)  and the intersection of the lines (blue circle).