Question 174867
{{{2(x+1)-(y-4)=15}}} Start with the first equation.



{{{2x+2-y+4=15}}} Distribute



{{{2x-y+6=15}}} Combine like terms.



{{{2x-y=15-6}}} Subtract 6 from both sides.



{{{2x-y=9}}} Combine like terms. 



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{{{3(x-1)+4(y+2)=2}}} Move onto the second equation.



{{{3x-3+4y+8=2}}} Distribute



{{{3x+4y+5=2}}} Combine like terms.



{{{3x+4y=2-5}}} Subtract 5 from both sides.



{{{3x+4y=-3}}} Combine like terms.




==================================================================







So we have the system of equations:


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



Let's solve this system by substitution



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


{{{2x-y=9}}} Start with the first equation



{{{-y=9-2x}}}  Subtract {{{2x}}} from both sides



{{{-y=-2x+9}}} Rearrange the equation



{{{y=(-2x+9)/(-1)}}} Divide both sides by {{{-1}}}



{{{y=((-2)/(-1))x+(9)/(-1)}}} Break up the fraction



{{{y=2x-9}}} Reduce




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Since {{{y=2x-9}}}, we can now replace each {{{y}}} in the second equation with {{{2x-9}}} to solve for {{{x}}}




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




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



{{{3x+8x-36=-3}}} Multiply



{{{11x-36=-3}}} Combine like terms on the left side



{{{11x=-3+36}}}Add 36 to both sides



{{{11x=33}}} Combine like terms on the right side



{{{x=(33)/(11)}}} Divide both sides by 11 to isolate x




{{{x=3}}} Divide






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



So the first part of our answer is: {{{x=3}}}










Since we know that {{{x=3}}} we can plug it into the equation {{{y=2x-9}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=2x-9}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=2(3)-9}}} Plug in {{{x=3}}}



{{{y=6-9}}} Multiply



{{{y=-3}}} Combine like terms 




-----------------Second Answer------------------------------



So the second part of our answer is: {{{y=-3}}}










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


So our answers are:


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


which form the ordered pair *[Tex \LARGE \left(3,-3\right)] 



This means that the system is consistent and independent.









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(3,-3\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (9-2*x)/(-1), (-3-3*x)/(4) ),
  blue(circle(3,-3,0.1)),
  blue(circle(3,-3,0.12)),
  blue(circle(3,-3,0.15))
)
}}} graph of {{{2x-y=9}}} (red) and {{{3x+4y=-3}}} (green)  and the intersection of the lines (blue circle).