Question 122678
Do you want to solve this using substitution?






Start with the given system of equations:


{{{system(2x-4y=7,4x+2y=-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


{{{2x-4y=7}}} Start with the first equation



{{{-4y=7-2x}}}  Subtract {{{2x}}} from both sides



{{{-4y=-2x+7}}} Rearrange the equation



{{{y=(-2x+7)/(-4)}}} Divide both sides by {{{-4}}}



{{{y=((-2)/(-4))x+(7)/(-4)}}} Break up the fraction



{{{y=(1/2)x-7/4}}} Reduce




Since {{{y=(1/2)x-7/4}}}, we can now replace each {{{y}}} in the second equation with {{{(1/2)x-7/4}}} to solve for {{{x}}}




{{{4x+2highlight(((1/2)x-7/4))=-1}}} For the second equation, replace {{{y}}} with {{{(1/2)x-7/4}}}. Since this eliminates {{{y}}}, we can now solve for {{{x}}}.




{{{4x+(2)(1/2)x+(2)(-7/4)=-1}}} Distribute {{{2}}} to {{{(1/2)x-7/4}}}



{{{4x+(2/2)x-14/4=-1}}} Multiply



{{{(4)(4x+(2/2)x-14/4)=(4)(-1)}}} Multiply both sides by the LCM of 4. 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>)




{{{16x+4x-14=-4}}} Distribute and multiply the LCM to each side




{{{20x-14=-4}}} Combine like terms on the left side



{{{20x=-4+14}}}Add 14 to both sides



{{{20x=10}}} Combine like terms on the right side



{{{x=(10)/(20)}}} Divide both sides by 20 to isolate x




{{{x=1/2}}} Reduce






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



So the first part of our answer is: {{{x=1/2}}}



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






Since we know that {{{x=1/2}}} we can plug it into the equation {{{y=(1/2)x-7/4}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=(1/2)x-7/4}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=(1/2)(1/2)-7/4}}} Plug in {{{x=1/2}}}



{{{y=1/4-7/4}}} Multiply



{{{y=-3/2}}} 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=-3/2}}}



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





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


So our answers are:


{{{x=1/2}}} and {{{y=-3/2}}}


which form the point *[Tex \LARGE \left(\frac{1}{2},-\frac{3}{2}\right)] 



which as an ordered pair, look like:





Notice if we graph the 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>)

{{{2x-4y=7}}}
{{{4x+2y=-1}}}

we get



{{{
drawing(500, 500, -10,10,-10,10,
  graph(500, 500, -10,10,-10,10, (7-2*x)/(-4), (-1-4*x)/(2) ),
  blue(circle(1/2,-3/2,0.1)),
  blue(circle(1/2,-3/2,0.12)),
  blue(circle(1/2,-3/2,0.15))
)
}}} graph of {{{2x-4y=7}}} (red) and {{{4x+2y=-1}}} (green)  and the intersection of the lines (blue circle).


and we can see that the two equations intersect at *[Tex \LARGE \left(\frac{1}{2},-\frac{3}{2}\right)]. This verifies our answer.