Question 252213



Start with the given system of equations:


{{{system(9x+5y=-6,4x+3y=2)}}}




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+5y=-6}}} Start with the first equation



{{{5y=-6-9x}}}  Subtract {{{9x}}} from both sides



{{{5y=-9x-6}}} Rearrange the equation



{{{y=(-9x-6)/(5)}}} Divide both sides by {{{5}}}



{{{y=((-9)/(5))x+(-6)/(5)}}} Break up the fraction



{{{y=(-9/5)x-6/5}}} Reduce




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


Since {{{y=(-9/5)x-6/5}}}, we can now replace each {{{y}}} in the second equation with {{{(-9/5)x-6/5}}} to solve for {{{x}}}




{{{4x+3highlight(((-9/5)x-6/5))=2}}} Plug in {{{y=(-9/5)x-6/5}}} into the second equation. In other words, replace each {{{y}}} with {{{(-9/5)x-6/5}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{4x+(3)(-9/5)x+(3)(-6/5)=2}}} Distribute {{{3}}} to {{{(-9/5)x-6/5}}}



{{{4x-(27/5)x-18/5=2}}} Multiply



{{{(5)(4x-(27/5)x-18/5)=(5)(2)}}} 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>)




{{{20x-27x-18=10}}} Distribute and multiply the LCM to each side




{{{-7x-18=10}}} Combine like terms on the left side



{{{-7x=10+18}}}Add 18 to both sides



{{{-7x=28}}} Combine like terms on the right side



{{{x=(28)/(-7)}}} Divide both sides by -7 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=(-9/5)x-6/5}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-9/5)x-6/5}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-9/5)(-4)-6/5}}} Plug in {{{x=-4}}}



{{{y=36/5-6/5}}} Multiply



{{{y=6}}} 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=6}}}










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


So our answers are:


{{{x=-4}}} and {{{y=6}}}


which form the point *[Tex \LARGE \left(-4,6\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(-4,6\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (-6-9*x)/(5), (2-4*x)/(3) ),
  blue(circle(-4,6,0.1)),
  blue(circle(-4,6,0.12)),
  blue(circle(-4,6,0.15))
)
}}} graph of {{{9x+5y=-6}}} (red) and {{{4x+3y=2}}} (green)  and the intersection of the lines (blue circle).