Question 201104


Start with the given system of equations:

{{{system(2x+y=4,x+4y=5)}}}



{{{-2(x+4y)=-2(5)}}} Multiply the both sides of the second equation by -2.



{{{-2x-8y=-10}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x+y=4,-2x-8y=-10)}}}



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+y)+(-2x-8y)=(4)+(-10)}}}



{{{(2x+-2x)+(1y+-8y)=4+-10}}} Group like terms.



{{{0x+-7y=-6}}} Combine like terms.



{{{-7y=-6}}} Simplify.



{{{y=(-6)/(-7)}}} Divide both sides by {{{-7}}} to isolate {{{y}}}.



{{{y=6/7}}} Reduce.



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



{{{2x+6/7=4}}} Plug in {{{y=6/7}}}.



{{{7(2x+6/cross(7))=7(4)}}} Multiply both sides by the LCD {{{7}}} to clear any fractions.



{{{14x+6=28}}} Distribute and multiply.



{{{14x=28-6}}} Subtract {{{6}}} from both sides.



{{{14x=22}}} Combine like terms on the right side.



{{{x=(22)/(14)}}} Divide both sides by {{{14}}} to isolate {{{x}}}.



{{{x=11/7}}} Reduce.



So the solutions are {{{x=11/7}}} and {{{y=6/7}}}.



Which form the ordered pair *[Tex \LARGE \left(\frac{11}{7},\frac{6}{7}\right)].



This means that the system is consistent and independent.