Question 581653


Start with the given system of equations:


{{{system(7x+y=51,8x-5y=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


{{{7x+y=51}}} Start with the first equation



{{{y=51-7x}}}  Subtract {{{7x}}} from both sides



{{{y=-7x+51}}} Rearrange the equation





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


Since {{{y=-7x+51}}}, we can now replace each {{{y}}} in the second equation with {{{-7x+51}}} to solve for {{{x}}}




{{{8x-5highlight((-7x+51))=3}}} Plug in {{{y=-7x+51}}} into the second equation. In other words, replace each {{{y}}} with {{{-7x+51}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{8x+(-5)(-7)x+(-5)(51)=3}}} Distribute {{{-5}}} to {{{-7x+51}}}



{{{8x+35x-255=3}}} Multiply



{{{43x-255=3}}} Combine like terms on the left side



{{{43x=3+255}}}Add 255 to both sides



{{{43x=258}}} Combine like terms on the right side



{{{x=(258)/(43)}}} Divide both sides by 43 to isolate x




{{{x=6}}} Divide






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



So the first part of our answer is: {{{x=6}}}










Since we know that {{{x=6}}} we can plug it into the equation {{{y=-7x+51}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-7x+51}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-7(6)+51}}} Plug in {{{x=6}}}



{{{y=-42+51}}} Multiply



{{{y=9}}} Combine like terms 




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



So the second part of our answer is: {{{y=9}}}










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


So our answers are:


{{{x=6}}} and {{{y=9}}}


which form the point *[Tex \LARGE \left(6,9\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(6,9\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (51-7*x)/(1), (3-8*x)/(-5) ),
  blue(circle(6,9,0.1)),
  blue(circle(6,9,0.12)),
  blue(circle(6,9,0.15))
)
}}} graph of {{{7x+y=51}}} (red) and {{{8x-5y=3}}} (green)  and the intersection of the lines (blue circle).