Question 124483
Start with the given system

{{{2x-5y=1}}}
{{{y=x-5}}}




{{{2x-5(x-5)=1}}}  Plug in {{{y=x-5}}} into the first equation. In other words, replace each {{{y}}} with {{{x-5}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.



{{{2x-5x+25=1}}} Distribute



{{{-3x+25=1}}} Combine like terms on the left side



{{{-3x=1-25}}}Subtract 25 from both sides



{{{-3x=-24}}} Combine like terms on the right side



{{{x=(-24)/(-3)}}} Divide both sides by -3 to isolate x




{{{x=8}}} Divide





Now that we know that {{{x=8}}}, we can plug this into {{{y=x-5}}} to find {{{y}}}




{{{y=(8)-5}}} Substitute {{{8}}} for each {{{x}}}



{{{y=3}}} Simplify



So our answer is {{{x=8}}} and {{{y=3}}} which also looks like *[Tex \LARGE \left(8,3\right)]



So because we got a unique solution, this means that the system is independent.



Notice if we graph the two equations, we can see that their intersection is at *[Tex \LARGE \left(8,3\right)]. So this verifies our answer.



{{{ graph( 500, 500, -5, 10, -5, 8, (1-2x)/(-5), y=x-5) }}} Graph of {{{2x-5y=1}}} (red) and {{{y=x-5}}} (green)