Question 126352



Start with the given system of equations:


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


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



{{{-2y=-5-5x}}}  Subtract {{{5x}}} from both sides



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



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



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



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




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


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




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




{{{(2)(-5x+(5/2)x+5/2)=(2)(3)}}} 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>)




{{{-10x+5x+5=6}}} Distribute and multiply the LCM to each side




{{{-5x+5=6}}} Combine like terms on the left side



{{{-5x=6-5}}}Subtract 5 from both sides



{{{-5x=1}}} Combine like terms on the right side



{{{x=(1)/(-5)}}} Divide both sides by -5 to isolate x




{{{x=-1/5}}} Reduce






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



So the first part of our answer is: {{{x=-1/5}}}










Since we know that {{{x=-1/5}}} we can plug it into the equation {{{y=(5/2)x+5/2}}} (remember we previously solved for {{{y}}} in the first equation).




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



{{{y=(5/2)(-1/5)+5/2}}} Plug in {{{x=-1/5}}}



{{{y=-5/10+5/2}}} Multiply



{{{y=2}}} 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=2}}}










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


So our answers are:


{{{x=-1/5}}} and {{{y=2}}}


which form the point *[Tex \LARGE \left(-\frac{1}{5},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(-\frac{1}{5},2\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-5-5*x)/(-2), (3--5*x)/(1) ),
  blue(circle(-1/5,2,0.1)),
  blue(circle(-1/5,2,0.12)),
  blue(circle(-1/5,2,0.15))
)
}}} graph of {{{5x-2y=-5}}} (red) and {{{-5x+y=3}}} (green)  and the intersection of the lines (blue circle).