Question 318610


Start with the given system of equations:

{{{system(14x-2y=78,2x-2y=6)}}}



{{{-1(2x-2y)=-1(6)}}} Multiply the both sides of the second equation by -1.



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



So we have the new system of equations:

{{{system(14x-2y=78,-2x+2y=-6)}}}



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



{{{(14x-2y)+(-2x+2y)=(78)+(-6)}}}



{{{(14x+-2x)+(-2y+2y)=78+-6}}} Group like terms.



{{{12x+0y=72}}} Combine like terms.



{{{12x=72}}} Simplify.



{{{x=(72)/(12)}}} Divide both sides by {{{12}}} to isolate {{{x}}}.



{{{x=6}}} Reduce.



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{{{14x-2y=78}}} Now go back to the first equation.



{{{14(6)-2y=78}}} Plug in {{{x=6}}}.



{{{84-2y=78}}} Multiply.



{{{-2y=78-84}}} Subtract {{{84}}} from both sides.



{{{-2y=-6}}} Combine like terms on the right side.



{{{y=(-6)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=3}}} Reduce.



So the solutions are {{{x=6}}} and {{{y=3}}}.



Which form the ordered pair *[Tex \LARGE \left(6,3\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(6,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-4,16,-7,13,
grid(1),
graph(500,500,-4,16,-7,13,(78-14x)/(-2),(6-2x)/(-2)),
circle(6,3,0.05),
circle(6,3,0.08),
circle(6,3,0.10)
)}}} Graph of {{{14x-2y=78}}} (red) and {{{2x-2y=6}}} (green)