Question 560785


Start with the given system of equations:

{{{system(-3x+y=7,5x+2y=3)}}}



{{{-2(-3x+y)=-2(7)}}} Multiply the both sides of the first equation by -2.



{{{6x-2y=-14}}} Distribute and multiply.



So we have the new system of equations:

{{{system(6x-2y=-14,5x+2y=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:



{{{(6x-2y)+(5x+2y)=(-14)+(3)}}}



{{{(6x+5x)+(-2y+2y)=-14+3}}} Group like terms.



{{{11x+0y=-11}}} Combine like terms.



{{{11x=-11}}} Simplify.



{{{x=(-11)/(11)}}} Divide both sides by {{{11}}} to isolate {{{x}}}.



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



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{{{6x-2y=-14}}} Now go back to the first equation.



{{{6(-1)-2y=-14}}} Plug in {{{x=-1}}}.



{{{-6-2y=-14}}} Multiply.



{{{-2y=-14+6}}} Add {{{6}}} to both sides.



{{{-2y=-8}}} Combine like terms on the right side.



{{{y=(-8)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=4}}} Reduce.



So the solutions are {{{x=-1}}} and {{{y=4}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,4\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(-1,4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-6,14,
grid(1),
graph(500,500,-11,9,-6,14,7+3x,(3-5x)/(2)),
circle(-1,4,0.05),
circle(-1,4,0.08),
circle(-1,4,0.10)
)}}} Graph of {{{-3x+y=7}}} (red) and {{{5x+2y=3}}} (green) 

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