Question 157501



Start with the given system of equations:


{{{system(9x+8y=-56,-2x+y=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


{{{9x+8y=-56}}} Start with the first equation



{{{8y=-56-9x}}}  Subtract {{{9x}}} from both sides



{{{8y=-9x-56}}} Rearrange the equation



{{{y=(-9x-56)/(8)}}} Divide both sides by {{{8}}}



{{{y=((-9)/(8))x+(-56)/(8)}}} Break up the fraction



{{{y=(-9/8)x-7}}} Reduce




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


Since {{{y=(-9/8)x-7}}}, we can now replace each {{{y}}} in the second equation with {{{(-9/8)x-7}}} to solve for {{{x}}}




{{{-2x+highlight(((-9/8)x-7))=18}}} Plug in {{{y=(-9/8)x-7}}} into the second equation. In other words, replace each {{{y}}} with {{{(-9/8)x-7}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{(8)(-2x-(9/8)x-7)=(8)(18)}}} Multiply both sides by the LCM of 8. 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>)




{{{-16x-9x-56=144}}} Distribute and multiply the LCM to each side




{{{-25x-56=144}}} Combine like terms on the left side



{{{-25x=144+56}}}Add 56 to both sides



{{{-25x=200}}} Combine like terms on the right side



{{{x=(200)/(-25)}}} Divide both sides by -25 to isolate x




{{{x=-8}}} Divide






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



So the first part of our answer is: {{{x=-8}}}










Since we know that {{{x=-8}}} we can plug it into the equation {{{y=(-9/8)x-7}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-9/8)x-7}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-9/8)(-8)-7}}} Plug in {{{x=-8}}}



{{{y=72/8-7}}} 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=-8}}} and {{{y=2}}}


which form the point *[Tex \LARGE \left(-8,2\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(-8,2\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-56-9*x)/(8), (18--2*x)/(1) ),
  blue(circle(-8,2,0.1)),
  blue(circle(-8,2,0.12)),
  blue(circle(-8,2,0.15))
)
}}} graph of {{{9x+8y=-56}}} (red) and {{{-2x+y=18}}} (green)  and the intersection of the lines (blue circle).