Question 165294


Start with the given system of equations:


{{{system(3x+y=7,4x-y=21)}}}




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=7}}} Start with the first equation



{{{y=7-3x}}}  Subtract {{{3x}}} from both sides



{{{y=-3x+7}}} Rearrange the equation



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


Since {{{y=-3x+7}}}, we can now replace each {{{y}}} in the second equation with {{{-3x+7}}} to solve for {{{x}}}




{{{4x-highlight((-3x+7))=21}}} Plug in {{{y=-3x+7}}} into the second equation. In other words, replace each {{{y}}} with {{{-3x+7}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{4x+3x-7=21}}} Distribute the negative



{{{7x-7=21}}} Combine like terms on the left side



{{{7x=21+7}}}Add 7 to both sides



{{{7x=28}}} Combine like terms on the right side



{{{x=(28)/(7)}}} Divide both sides by 7 to isolate x




{{{x=4}}} Divide






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



So the first part of our answer is: {{{x=4}}}










Since we know that {{{x=4}}} we can plug it into the equation {{{y=-3x+7}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-3x+7}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-3(4)+7}}} Plug in {{{x=4}}}



{{{y=-12+7}}} Multiply



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




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



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










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


So our answers are:


{{{x=4}}} and {{{y=-5}}}


which form the point *[Tex \LARGE \left(4,-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(4,-5\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (7-3*x)/(1), (21-4*x)/(-1) ),
  blue(circle(4,-5,0.1)),
  blue(circle(4,-5,0.12)),
  blue(circle(4,-5,0.15))
)
}}} graph of {{{3x+y=7}}} (red) and {{{4x-y=21}}} (green)  and the intersection of the lines (blue circle).