Question 325295



Start with the given system of equations:


{{{system(-3x-4y=2,3x+3y=-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


{{{-3x-4y=2}}} Start with the first equation



{{{-4y=2+3x}}} Add {{{3x}}} to both sides



{{{-4y=3x+2}}} Rearrange the equation



{{{y=(3x+2)/(-4)}}} Divide both sides by {{{-4}}}



{{{y=((3)/(-4))x+(2)/(-4)}}} Break up the fraction



{{{y=(-3/4)x-1/2}}} Reduce




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


Since {{{y=(-3/4)x-1/2}}}, we can now replace each {{{y}}} in the second equation with {{{(-3/4)x-1/2}}} to solve for {{{x}}}




{{{3x+3highlight(((-3/4)x-1/2))=-3}}} Plug in {{{y=(-3/4)x-1/2}}} into the second equation. In other words, replace each {{{y}}} with {{{(-3/4)x-1/2}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{3x+(3)(-3/4)x+(3)(-1/2)=-3}}} Distribute {{{3}}} to {{{(-3/4)x-1/2}}}



{{{3x-(9/4)x-3/2=-3}}} Multiply



{{{(4)(3x-(9/4)x-3/2)=(4)(-3)}}} 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>)




{{{12x-9x-6=-12}}} Distribute and multiply the LCM to each side




{{{3x-6=-12}}} Combine like terms on the left side



{{{3x=-12+6}}}Add 6 to both sides



{{{3x=-6}}} Combine like terms on the right side



{{{x=(-6)/(3)}}} Divide both sides by 3 to isolate x




{{{x=-2}}} Divide






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



So the first part of our answer is: {{{x=-2}}}










Since we know that {{{x=-2}}} we can plug it into the equation {{{y=(-3/4)x-1/2}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-3/4)x-1/2}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-3/4)(-2)-1/2}}} Plug in {{{x=-2}}}



{{{y=6/4-1/2}}} Multiply



{{{y=1}}} 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=1}}}










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


So our answers are:


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


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





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (2--3*x)/(-4), (-3-3*x)/(3) ),
  blue(circle(-2,1,0.1)),
  blue(circle(-2,1,0.12)),
  blue(circle(-2,1,0.15))
)
}}} graph of {{{-3x-4y=2}}} (red) and {{{3x+3y=-3}}} (green)  and the intersection of the lines (blue circle).