Question 227942


Start with the given system of equations:

{{{system(2x+5y=1,-x+6y=8)}}}



{{{2(-x+6y)=2(8)}}} Multiply the both sides of the second equation by 2.



{{{-2x+12y=16}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x+5y=1,-2x+12y=16)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(2x+5y)+(-2x+12y)=(1)+(16)}}}



{{{(2x+-2x)+(5y+12y)=1+16}}} Group like terms.



{{{0x+17y=17}}} Combine like terms.



{{{17y=17}}} Simplify.



{{{y=(17)/(17)}}} Divide both sides by {{{17}}} to isolate {{{y}}}.



{{{y=1}}} Reduce.



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{{{2x+5y=1}}} Now go back to the first equation.



{{{2x+5(1)=1}}} Plug in {{{y=1}}}.



{{{2x+5=1}}} Multiply.



{{{2x=1-5}}} Subtract {{{5}}} from both sides.



{{{2x=-4}}} Combine like terms on the right side.



{{{x=(-4)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}.



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



So the solutions are {{{x=-2}}} and {{{y=1}}}.



Which form the ordered pair *[Tex \LARGE \left(-2,1\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-2,1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-12,8,-9,11,
grid(1),
graph(500,500,-12,8,-9,11,(1-2x)/(5),(8+x)/(6)),
circle(-2,1,0.05),
circle(-2,1,0.08),
circle(-2,1,0.10)
)}}} Graph of {{{2x+5y=1}}} (red) and {{{-x+6y=8}}} (green)