Question 253263


Start with the given system of equations:

{{{system(x+2y=1,2x+5y=3)}}}



{{{-2(x+2y)=-2(1)}}} Multiply the both sides of the first equation by -2.



{{{-2x-4y=-2}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-2x-4y=-2,2x+5y=3)}}}



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-4y)+(2x+5y)=(-2)+(3)}}}



{{{(-2x+2x)+(-4y+5y)=-2+3}}} Group like terms.



{{{0x+y=1}}} Combine like terms.



{{{y=1}}} Simplify.



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{{{-2x-4y=-2}}} Now go back to the first equation.



{{{-2x-4(1)=-2}}} Plug in {{{y=1}}}.



{{{-2x-4=-2}}} Multiply.



{{{-2x=-2+4}}} Add {{{4}}} to both sides.



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



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



{{{x=-1}}} Reduce.



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



Which form the ordered pair *[Tex \LARGE \left(-1,1\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(-1,1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-9,11,
grid(1),
graph(500,500,-11,9,-9,11,(1-x)/(2),(3-2x)/(5)),
circle(-1,1,0.05),
circle(-1,1,0.08),
circle(-1,1,0.10)
)}}} Graph of {{{x+2y=1}}} (red) and {{{2x+5y=3}}} (green)