Question 202419


Start with the given system of equations:



{{{system(4x+5y=8,-6x+y=56)}}}



{{{4x+5y=8}}} Start with the first equation.



{{{5y=8-4x}}} Subtract {{{4x}}} from both sides.



{{{y=(8-4x)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=-(4/5)x+8/5}}} Rearrange the terms and simplify.



-------------------------------------------



{{{-6x+y=56}}} Move onto the second equation.



{{{-6x-(4/5)x+8/5=56}}} Now plug in {{{y=-(4/5)x+8/5}}}.



{{{5(-6x)-cross(5)((4/cross(5))x)+cross(5)(8/cross(5))=5(56)}}} Multiply EVERY term by the LCD {{{5}}} to clear any fractions.



{{{-30x-4x+8=280}}} Multiply and simplify.



{{{-34x+8=280}}} Combine like terms on the left side.



{{{-34x=280-8}}} Subtract {{{8}}} from both sides.



{{{-34x=272}}} Combine like terms on the right side.



{{{x=(272)/(-34)}}} Divide both sides by {{{-34}}} to isolate {{{x}}}.



{{{x=-8}}} Reduce.



-------------------------------------------



Since we know that {{{x=-8}}}, we can use this to find {{{y}}}.



{{{4x+5y=8}}} Go back to the first equation.



{{{4(-8)+5y=8}}} Plug in {{{x=-8}}}.



{{{-32+5y=8}}} Multiply.



{{{5y=8+32}}} Add {{{32}}} to both sides.



{{{5y=40}}} Combine like terms on the right side.



{{{y=(40)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.



{{{y=8}}} Reduce.



So the solutions are {{{x=-8}}} and {{{y=8}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-8,8\right)]. So this visually verifies our answer.



{{{drawing(500,500,-18,2,-2,18,
grid(1),
graph(500,500,-18,2,-2,18,(8-4x)/(5),56+6x),
circle(-8,8,0.05),
circle(-8,8,0.08),
circle(-8,8,0.10)
)}}} Graph of {{{4x+5y=8}}} (red) and {{{-6x+y=56}}} (green)