Question 126403



Start with the given system of equations:


{{{system(x+y=13,3x+4y=44)}}}




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=13}}} Start with the first equation



{{{y=13-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+13}}} Rearrange the equation



{{{y=(-x+13)/(1)}}} Divide both sides by {{{1}}}



{{{y=((-1)/(1))x+(13)/(1)}}} Break up the fraction



{{{y=-x+13}}} Reduce




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


Since {{{y=-x+13}}}, we can now replace each {{{y}}} in the second equation with {{{-x+13}}} to solve for {{{x}}}




{{{3x+4highlight((-x+13))=44}}} Plug in {{{y=-x+13}}} into the first equation. In other words, replace each {{{y}}} with {{{-x+13}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{3x+(4)(-1)x+(4)(13)=44}}} Distribute {{{4}}} to {{{-x+13}}}



{{{3x-4x+52=44}}} Multiply



{{{-x+52=44}}} Combine like terms on the left side



{{{-x=44-52}}}Subtract 52 from both sides



{{{-x=-8}}} Combine like terms on the right side



{{{x=(-8)/(-1)}}} Divide both sides by -1 to isolate x




{{{x=8}}} Divide






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



So the first part of our answer is: {{{x=8}}}










Since we know that {{{x=8}}} we can plug it into the equation {{{y=-x+13}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+13}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(8)+13}}} Plug in {{{x=8}}}



{{{y=-8+13}}} Multiply



{{{y=5}}} Combine like terms 




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



So the second part of our answer is: {{{y=5}}}










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


So our answers are:


{{{x=8}}} and {{{y=5}}}


which form the point *[Tex \LARGE \left(8,5\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(8,5\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (13-1*x)/(1), (44-3*x)/(4) ),
  blue(circle(8,5,0.1)),
  blue(circle(8,5,0.12)),
  blue(circle(8,5,0.15))
)
}}} graph of {{{x+y=13}}} (red) and {{{3x+4y=44}}} (green)  and the intersection of the lines (blue circle).