Question 145832



Start with the given system of equations:


{{{system(3x+5y=-11,-6x+y=44)}}}




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+5y=-11}}} Start with the first equation



{{{5y=-11-3x}}}  Subtract {{{3x}}} from both sides



{{{5y=-3x-11}}} Rearrange the equation



{{{y=(-3x-11)/(5)}}} Divide both sides by {{{5}}}



{{{y=((-3)/(5))x+(-11)/(5)}}} Break up the fraction



{{{y=(-3/5)x-11/5}}} Reduce




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


Since {{{y=(-3/5)x-11/5}}}, we can now replace each {{{y}}} in the second equation with {{{(-3/5)x-11/5}}} to solve for {{{x}}}




{{{-6x+highlight(((-3/5)x-11/5))=44}}} Plug in {{{y=(-3/5)x-11/5}}} into the first equation. In other words, replace each {{{y}}} with {{{(-3/5)x-11/5}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{(5)(-6x-(3/5)x-11/5)=(5)(44)}}} Multiply both sides by the LCM of 5. 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>)




{{{-30x-3x-11=220}}} Distribute and multiply the LCM to each side




{{{-33x-11=220}}} Combine like terms on the left side



{{{-33x=220+11}}}Add 11 to both sides



{{{-33x=231}}} Combine like terms on the right side



{{{x=(231)/(-33)}}} Divide both sides by -33 to isolate x




{{{x=-7}}} Divide






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



So the first part of our answer is: {{{x=-7}}}










Since we know that {{{x=-7}}} we can plug it into the equation {{{y=(-3/5)x-11/5}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-3/5)x-11/5}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-3/5)(-7)-11/5}}} Plug in {{{x=-7}}}



{{{y=21/5-11/5}}} 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=-7}}} and {{{y=2}}}


which form the point *[Tex \LARGE \left(-7,2\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(-7,2\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-11-3*x)/(5), (44--6*x)/(1) ),
  blue(circle(-7,2,0.1)),
  blue(circle(-7,2,0.12)),
  blue(circle(-7,2,0.15))
)
}}} graph of {{{3x+5y=-11}}} (red) and {{{-6x+y=44}}} (green)  and the intersection of the lines (blue circle).