Question 202420


Start with the given system of equations:



{{{system(3x+8y=66,-3x+y=15)}}}



{{{3x+8y=66}}} Start with the first equation.



{{{8y=66-3x}}} Subtract {{{3x}}} from both sides.



{{{y=(66-3x)/(8)}}} Divide both sides by {{{8}}} to isolate {{{y}}}.



{{{y=-(3/8)x+33/4}}} Rearrange the terms and simplify.



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{{{-3x+y=15}}} Move onto the second equation.



{{{-3x-(3/8)x+33/4=15}}} Now plug in {{{y=-(3/8)x+33/4}}}.



{{{8(-3x)-cross(8)((3/cross(8))x)+cross(8)^2(33/cross(4))=8(15)}}} Multiply EVERY term by the LCD {{{8}}} to clear any fractions.



{{{-24x-3x+66=120}}} Multiply and simplify.



{{{-27x+66=120}}} Combine like terms on the left side.



{{{-27x=120-66}}} Subtract {{{66}}} from both sides.



{{{-27x=54}}} Combine like terms on the right side.



{{{x=(54)/(-27)}}} Divide both sides by {{{-27}}} to isolate {{{x}}}.



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



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Since we know that {{{x=-2}}}, we can use this to find {{{y}}}.



{{{3x+8y=66}}} Go back to the first equation.



{{{3(-2)+8y=66}}} Plug in {{{x=-2}}}.



{{{-6+8y=66}}} Multiply.



{{{8y=66+6}}} Add {{{6}}} to both sides.



{{{8y=72}}} Combine like terms on the right side.



{{{y=(72)/(8)}}} Divide both sides by {{{8}}} to isolate {{{y}}}.



{{{y=9}}} Reduce.



So the solutions are {{{x=-2}}} and {{{y=9}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-2,9\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-1,19,
grid(1),
graph(500,500,-12,8,-1,19,(66-3x)/(8),15+3x),
circle(-2,9,0.05),
circle(-2,9,0.08),
circle(-2,9,0.10)
)}}} Graph of {{{3x+8y=66}}} (red) and {{{-3x+1y=15}}} (green)