Question 186754


Start with the given system of equations:

{{{system(2x-3y=7,3x+7y=-1)}}}



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



{{{6x-9y=21}}} Distribute and multiply.



{{{-2(3x+7y)=-2(-1)}}} Multiply the both sides of the second equation by -2.



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



So we have the new system of equations:

{{{system(6x-9y=21,-6x-14y=2)}}}



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-9y)+(-6x-14y)=(21)+(2)}}}



{{{(6x+-6x)+(-9y+-14y)=21+2}}} Group like terms.



{{{0x+-23y=23}}} Combine like terms.



{{{-23y=23}}} Simplify.



{{{y=(23)/(-23)}}} Divide both sides by {{{-23}}} to isolate {{{y}}}.



{{{y=-1}}} Reduce.



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{{{6x-9y=21}}} Now go back to the first equation.



{{{6x-9(-1)=21}}} Plug in {{{y=-1}}}.



{{{6x+9=21}}} Multiply.



{{{6x=21-9}}} Subtract {{{9}}} from both sides.



{{{6x=12}}} Combine like terms on the right side.



{{{x=(12)/(6)}}} Divide both sides by {{{6}}} to isolate {{{x}}}.



{{{x=2}}} Reduce.



So the solutions are {{{x=2}}} and {{{y=-1}}}.



Which form the ordered pair *[Tex \LARGE \left(2,-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(2,-1\right)]. So this visually verifies our answer.



{{{drawing(500,500,-8,12,-11,9,
grid(1),
graph(500,500,-8,12,-11,9,(7-2x)/(-3),(-1-3x)/(7)),
circle(2,-1,0.05),
circle(2,-1,0.08),
circle(2,-1,0.10)
)}}} Graph of {{{2x-3y=7}}} (red) and {{{3x+7y=-1}}} (green)