Question 241743
Start with the given system of equations:

{{{system(3x+y=7,4x+2y=16)}}}



{{{-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=-14,4x+2y=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:



{{{(-6x-2y)+(4x+2y)=(-14)+(16)}}}



{{{(-6x+4x)+(-2y+2y)=-14+16}}} Group like terms.



{{{-2x+0y=2}}} Combine like terms.



{{{-2x=2}}} Simplify.



{{{x=(2)/(-2)}}} Divide both sides by {{{-2}}} 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}}} Subtract {{{6}}} from both sides.



{{{-2y=-20}}} Combine like terms on the right side.



{{{y=(-20)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=10}}} Reduce.



So the solutions are {{{x=-1}}} and {{{y=10}}}.



Which form the ordered pair *[Tex \LARGE \left(-1,10\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(-1,10\right)]. So this visually verifies our answer.



{{{drawing(500,500,-11,9,-5,15,
grid(1),
graph(500,500,-11,9,-5,15,7-3x,(16-4x)/(2)),
circle(-1,10,0.05),
circle(-1,10,0.08),
circle(-1,10,0.10)
)}}} Graph of {{{3x+y=7}}} (red) and {{{4x+2y=16}}} (green)