Question 201434

Start with the given system of equations:

{{{system(2x+3y=4,x+4y=6)}}}



{{{-2(x+4y)=-2(6)}}} Multiply the both sides of the second equation by -2.



{{{-2x-8y=-12}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x+3y=4,-2x-8y=-12)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(2x+3y)+(-2x-8y)=(4)+(-12)}}}



{{{(2x-2x)+(3y-8y)=4-12}}} Group like terms.



{{{0x-5y=-8}}} Combine like terms.



{{{-5y=-8}}} Simplify.



{{{y=(-8)/(-5)}}} Divide both sides by {{{-5}}} to isolate {{{y}}}.



{{{y=8/5}}} Reduce.



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



{{{2x+3y=4}}} Now go back to the first equation.



{{{2x+3(8/5)=4}}} Plug in {{{y=8/5}}}.



{{{2x+24/5=4}}} Multiply.



{{{5(2x+24/cross(5))=5(4)}}} Multiply both sides by the LCD {{{5}}} to clear any fractions.



{{{10x+24=20}}} Distribute and multiply.



{{{10x=20-24}}} Subtract 24 from both sides.



{{{10x=-4}}} Combine like terms.



{{{x=(-4)/(10)}}} Divide both sides by {{{10}}} to isolate {{{x}}}.



{{{x=-2/5}}} Reduce.



So the solutions are {{{x=-2/5}}} and {{{y=8/5}}}.



Which form the ordered pair *[Tex \LARGE \left(-\frac{2}{5},\frac{8}{5}\right)].



This means that the system is consistent and independent.