Question 376337


Start with the given system of equations:

{{{system(x-y=-1,5x+2y=30)}}}



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



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



So we have the new system of equations:

{{{system(2x-2y=-2,5x+2y=30)}}}



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-2y)+(5x+2y)=(-2)+(30)}}}



{{{(2x+5x)+(-2y+2y)=-2+30}}} Group like terms.



{{{7x+0y=28}}} Combine like terms.



{{{7x=28}}} Simplify.



{{{x=(28)/(7)}}} Divide both sides by {{{7}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



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



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



{{{8-2y=-2}}} Multiply.



{{{-2y=-2-8}}} Subtract {{{8}}} from both sides.



{{{-2y=-10}}} Combine like terms on the right side.



{{{y=(-10)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=5}}} Reduce.



So the solutions are {{{x=4}}} and {{{y=5}}}.



Which form the ordered pair *[Tex \LARGE \left(4,5\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(4,5\right)]. So this visually verifies our answer.



{{{drawing(500,500,-6,14,-5,15,
grid(1),
graph(500,500,-6,14,-5,15,(-1-x)/(-1),(30-5x)/(2)),
circle(4,5,0.05),
circle(4,5,0.08),
circle(4,5,0.10)
)}}} Graph of {{{x-y=-1}}} (red) and {{{5x+2y=30}}} (green) 



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