Question 124955



Start with the given system of equations:


{{{system(x+3y=2,-x+y=1)}}}




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+3y=2}}} Start with the first equation



{{{3y=2-x}}}  Subtract {{{x}}} from both sides



{{{3y=-x+2}}} Rearrange the equation



{{{y=(-x+2)/(3)}}} Divide both sides by {{{3}}}



{{{y=((-1)/(3))x+(2)/(3)}}} Break up the fraction



{{{y=(-1/3)x+2/3}}} Reduce




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


Since {{{y=(-1/3)x+2/3}}}, we can now replace each {{{y}}} in the second equation with {{{(-1/3)x+2/3}}} to solve for {{{x}}}




{{{-x+highlight(((-1/3)x+2/3))=1}}} Plug in {{{y=(-1/3)x+2/3}}} into the first equation. In other words, replace each {{{y}}} with {{{(-1/3)x+2/3}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{(3)(-1x-(1/3)x+2/3)=(3)(1)}}} Multiply both sides by the LCM of 3. This will eliminate the fractions  (note: if you need help with finding the LCM, check out this <a href=http://www.algebra.com/algebra/homework/divisibility/least-common-multiple.solver>solver</a>)




{{{-3x-1x+2=3}}} Distribute and multiply the LCM to each side




{{{-4x+2=3}}} Combine like terms on the left side



{{{-4x=3-2}}}Subtract 2 from both sides



{{{-4x=1}}} Combine like terms on the right side



{{{x=(1)/(-4)}}} Divide both sides by -4 to isolate x




{{{x=-1/4}}} Reduce






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



So the first part of our answer is: {{{x=-1/4}}}










Since we know that {{{x=-1/4}}} we can plug it into the equation {{{y=(-1/3)x+2/3}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(-1/3)x+2/3}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(-1/3)(-1/4)+2/3}}} Plug in {{{x=-1/4}}}



{{{y=1/12+2/3}}} Multiply



{{{y=3/4}}} Combine like terms and reduce.  (note: if you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>)




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



So the second part of our answer is: {{{y=3/4}}}










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


So our answers are:


{{{x=-1/4}}} and {{{y=3/4}}}


which form the point *[Tex \LARGE \left(-\frac{1}{4},\frac{3}{4}\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(-\frac{1}{4},\frac{3}{4}\right)]. This visually verifies our answer.





{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (2-1*x)/(3), (1--1*x)/(1) ),
  blue(circle(-1/4,3/4,0.1)),
  blue(circle(-1/4,3/4,0.12)),
  blue(circle(-1/4,3/4,0.15))
)
}}} graph of {{{x+3y=2}}} (red) and {{{-x+y=1}}} (green)  and the intersection of the lines (blue circle).