Question 552815



Start with the given system of equations:


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




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


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



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



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




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


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




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




{{{2x+(3)(-1)x+(3)(4)=9}}} Distribute {{{3}}} to {{{-x+4}}}



{{{2x-3x+12=9}}} Multiply



{{{-x+12=9}}} Combine like terms on the left side



{{{-x=9-12}}}Subtract 12 from both sides



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



{{{x=(-3)/(-1)}}} Divide both sides by -1 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=-x+4}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+4}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(3)+4}}} Plug in {{{x=3}}}



{{{y=-3+4}}} Multiply



{{{y=1}}} Combine like terms 




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



So the second part of our answer is: {{{y=1}}}










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


So our answers are:


{{{x=3}}} and {{{y=1}}}


which form the point *[Tex \LARGE \left(3,1\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(3,1\right)]. This visually verifies our answer.





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



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