Question 319710


Start with the given system of equations:

{{{system(2x-4y=6,4x+y=21)}}}



{{{4(4x+y)=4(21)}}} Multiply the both sides of the second equation by 4.



{{{16x+4y=84}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x-4y=6,16x+4y=84)}}}



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-4y)+(16x+4y)=(6)+(84)}}}



{{{(2x+16x)+(-4y+4y)=6+84}}} Group like terms.



{{{18x+0y=90}}} Combine like terms.



{{{18x=90}}} Simplify.



{{{x=(90)/(18)}}} Divide both sides by {{{18}}} to isolate {{{x}}}.



{{{x=5}}} Reduce.



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{{{2x-4y=6}}} Now go back to the first equation.



{{{2(5)-4y=6}}} Plug in {{{x=5}}}.



{{{10-4y=6}}} Multiply.



{{{-4y=6-10}}} Subtract {{{10}}} from both sides.



{{{-4y=-4}}} Combine like terms on the right side.



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



{{{y=1}}} Reduce.



So the solutions are {{{x=5}}} and {{{y=1}}}.



Which form the ordered pair *[Tex \LARGE \left(5,1\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(5,1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-5,15,-9,11,
grid(1),
graph(500,500,-5,15,-9,11,(6-2x)/(-4),21-4x),
circle(5,1,0.05),
circle(5,1,0.08),
circle(5,1,0.10)
)}}} Graph of {{{2x-4y=6}}} (red) and {{{4x+y=21}}} (green)