Question 215115


Start with the given system of equations:



{{{system(4x+3y=-11,-8x+y=43)}}}



{{{-8x+y=43}}} Start with the second equation.



{{{y=43+8x}}} Add {{{8x}}} to both sides.



{{{y=8x+43}}} Rearrange the terms and simplify.



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{{{4x+3y=-11}}} Move back to the first equation.



{{{4x+3(8x+43)=-11}}} Plug in {{{y=8x+43}}}.



{{{4x+24x+129=-11}}} Distribute.



{{{28x+129=-11}}} Combine like terms on the left side.



{{{28x=-11-129}}} Subtract {{{129}}} from both sides.



{{{28x=-140}}} Combine like terms on the right side.



{{{x=(-140)/(28)}}} Divide both sides by {{{28}}} to isolate {{{x}}}.



{{{x=-5}}} Reduce.



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Since we know that {{{x=-5}}}, we can use this to find {{{y}}}.



{{{-8x+y=43}}} Move onto the second equation.



{{{-8(-5)+y=43}}} Plug in {{{x=-5}}}.



{{{40+y=43}}} Multiply.



{{{y=43-40}}} Subtract {{{40}}} from both sides.



{{{y=3}}} Combine like terms on the right side.



So the solutions are {{{x=-5}}} and {{{y=3}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-5,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-15,5,-7,13,
grid(1),
graph(500,500,-15,5,-7,13,(-11-4x)/(3),43+8x),
circle(-5,3,0.05),
circle(-5,3,0.08),
circle(-5,3,0.10)
)}}} Graph of {{{4x+3y=-11}}} (red) and {{{-8x+y=43}}} (green)