Question 590780


Start with the given system of equations:

{{{system(3x-2y=11,6x+11y=97)}}}



{{{-2(3x-2y)=-2(11)}}} Multiply the both sides of the first equation by -2.



{{{-6x+4y=-22}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-6x+4y=-22,6x+11y=97)}}}



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



{{{(-6x+4y)+(6x+11y)=(-22)+(97)}}}



{{{(-6x+6x)+(4y+11y)=-22+97}}} Group like terms.



{{{0x+15y=75}}} Combine like terms.



{{{15y=75}}} Simplify.



{{{y=(75)/(15)}}} Divide both sides by {{{15}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



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{{{-6x+4y=-22}}} Now go back to the first equation.



{{{-6x+4(5)=-22}}} Plug in {{{y=5}}}.



{{{-6x+20=-22}}} Multiply.



{{{-6x=-22-20}}} Subtract {{{20}}} from both sides.



{{{-6x=-42}}} Combine like terms on the right side.



{{{x=(-42)/(-6)}}} Divide both sides by {{{-6}}} to isolate {{{x}}}.



{{{x=7}}} Reduce.



So the solutions are {{{x=7}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(7,5\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(7,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-3,17,-5,15,
grid(1),
graph(500,500,-3,17,-5,15,(11-3x)/(-2),(97-6x)/(11)),
circle(7,5,0.05),
circle(7,5,0.08),
circle(7,5,0.10)
)}}} Graph of {{{3x-2y=11}}} (red) and {{{6x+11y=97}}} (green) 


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