Question 161096



Start with the given system of equations:


{{{system(7x+4y=-16,-9x+y=39)}}}




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


{{{7x+4y=-16}}} Start with the first equation



{{{4y=-16-7x}}}  Subtract {{{7x}}} from both sides



{{{4y=-7x-16}}} Rearrange the equation



{{{y=(-7x-16)/(4)}}} Divide both sides by {{{4}}}



{{{y=((-7)/(4))x+(-16)/(4)}}} Break up the fraction



{{{y=(-7/4)x-4}}} Reduce




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


Since {{{y=(-7/4)x-4}}}, we can now replace each {{{y}}} in the second equation with {{{(-7/4)x-4}}} to solve for {{{x}}}




{{{-9x+highlight(((-7/4)x-4))=39}}} Plug in {{{y=(-7/4)x-4}}} into the second equation. In other words, replace each {{{y}}} with {{{(-7/4)x-4}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{(4)(-9x-(7/4)x-4)=(4)(39)}}} Multiply both sides by the LCM of 4. 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>)




{{{-36x-7x-16=156}}} Distribute and multiply the LCM to each side




{{{-43x-16=156}}} Combine like terms on the left side



{{{-43x=156+16}}}Add 16 to both sides



{{{-43x=172}}} Combine like terms on the right side



{{{x=(172)/(-43)}}} Divide both sides by -43 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=(-7/4)x-4}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-7/4)x-4}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-7/4)(-4)-4}}} Plug in {{{x=-4}}}



{{{y=28/4-4}}} Multiply



{{{y=3}}} 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=3}}}










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


So our answers are:


{{{x=-4}}} and {{{y=3}}}


which form the point *[Tex \LARGE \left(-4,3\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(-4,3\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-16-7*x)/(4), (39--9*x)/(1) ),
  blue(circle(-4,3,0.1)),
  blue(circle(-4,3,0.12)),
  blue(circle(-4,3,0.15))
)
}}} graph of {{{7x+4y=-16}}} (red) and {{{-9x+y=39}}} (green)  and the intersection of the lines (blue circle).