Question 395356



Start with the given system of equations:


{{{system(-3x+y=4,x+3y=22)}}}




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



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



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



{{{y=(+3x+4)/(1)}}} Divide both sides by {{{1}}}



{{{y=((+3)/(1))x+(4)/(1)}}} Break up the fraction



{{{y=3x+4}}} Reduce




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


Since {{{y=3x+4}}}, we can now replace each {{{y}}} in the second equation with {{{3x+4}}} to solve for {{{x}}}




{{{x+3highlight((3x+4))=22}}} Plug in {{{y=3x+4}}} into the second equation. In other words, replace each {{{y}}} with {{{3x+4}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{x+(3)(3)x+(3)(4)=22}}} Distribute {{{3}}} to {{{3x+4}}}



{{{x+9x+12=22}}} Multiply



{{{10x+12=22}}} Combine like terms on the left side



{{{10x=22-12}}}Subtract 12 from both sides



{{{10x=10}}} Combine like terms on the right side



{{{x=(10)/(10)}}} Divide both sides by 10 to isolate x




{{{x=1}}} Divide






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



So the first part of our answer is: {{{x=1}}}










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




{{{y=3x+4}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=3(1)+4}}} Plug in {{{x=1}}}



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



{{{y=7}}} Combine like terms 




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



So the second part of our answer is: {{{y=7}}}










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


So our answers are:


{{{x=1}}} and {{{y=7}}}


which form the point *[Tex \LARGE \left(1,7\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(1,7\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (4--3*x)/(1), (22-1*x)/(3) ),
  blue(circle(1,7,0.1)),
  blue(circle(1,7,0.12)),
  blue(circle(1,7,0.15))
)
}}} graph of {{{-3x+y=4}}} (red) and {{{x+3y=22}}} (green)  and the intersection of the lines (blue circle).




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Jim