Question 202188

Start with the given system of equations:



{{{system(-3x+5y=1,2x+4y=80)}}}



{{{-3x+5y=1}}} Start with the first equation.



{{{5y=1+3x}}} Add {{{3x}}} to both sides.



{{{y=(1+3x)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=(3/5)x+1/5}}} Rearrange the terms and simplify.



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{{{2x+4y=80}}} Move onto the second equation.



{{{2x+4((3/5)x+1/5)=80}}} Now plug in {{{y=(3/5)x+1/5}}}.



{{{2x+(12/5)x+4/5=80}}} Distribute.



{{{5(2x+(12/cross(5))x+4/cross(5))=5(80)}}} Multiply both sides by the LCD {{{5}}} to clear any fractions.



{{{10x+12x+4=400}}} Distribute and multiply.



{{{22x+4=400}}} Combine like terms on the left side.



{{{22x=400-4}}} Subtract {{{4}}} from both sides.



{{{22x=396}}} Combine like terms on the right side.



{{{x=(396)/(22)}}} Divide both sides by {{{22}}} to isolate {{{x}}}.



{{{x=18}}} Reduce.



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Since we know that {{{x=18}}}, we can use this to find {{{y}}}.



{{{-3x+5y=1}}} Go back to the first equation.



{{{-3(18)+5y=1}}} Plug in {{{x=18}}}.



{{{-54+5y=1}}} Multiply.



{{{5y=1+54}}} Add {{{54}}} to both sides.



{{{5y=55}}} Combine like terms on the right side.



{{{y=(55)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=11}}} Reduce.



So the solutions are {{{x=18}}} and {{{y=11}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(18,11\right)]. So this visually verifies our answer.



{{{drawing(500,500,-2,20,-5,15,
grid(1),
graph(500,500,-2,20,-5,15,(1+3x)/(5),(80-2x)/(4)),
circle(18,11,0.05),
circle(18,11,0.08),
circle(18,11,0.10)
)}}} Graph of {{{-3x+5y=1}}} (red) and {{{2x+4y=80}}} (green)