Question 124909



Start with the given system of equations:


{{{system(3x-y=-7,x+y=-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


{{{3x-y=-7}}} Start with the first equation



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



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



{{{y=(-3x-7)/(-1)}}} Divide both sides by {{{-1}}}



{{{y=((-3)/(-1))x+(-7)/(-1)}}} Break up the fraction



{{{y=3x+7}}} Reduce




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


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




{{{x+highlight((3x+7))=-9}}} Plug in {{{y=3x+7}}} into the first 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+7=-9}}} Combine like terms on the left side



{{{4x=-9-7}}}Subtract 7 from both sides



{{{4x=-16}}} Combine like terms on the right side



{{{x=(-16)/(4)}}} Divide both sides by 4 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), (-9-1*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 {{{x+y=-9}}} (green)  and the intersection of the lines (blue circle).