Question 202430


Start with the given system of equations:

{{{system(7x-6y=13,6x-5y=11)}}}



{{{5(7x-6y)=5(13)}}} Multiply the both sides of the first equation by 5.



{{{35x-30y=65}}} Distribute and multiply.



{{{-6(6x-5y)=-6(11)}}} Multiply the both sides of the second equation by -6.



{{{-36x+30y=-66}}} Distribute and multiply.



So we have the new system of equations:

{{{system(35x-30y=65,-36x+30y=-66)}}}



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



{{{(35x-30y)+(-36x+30y)=(65)+(-66)}}}



{{{(35x+-36x)+(-30y+30y)=65+-66}}} Group like terms.



{{{-x+0y=-1}}} Combine like terms.



{{{-x=-1}}} Simplify.



{{{x=(-1)/(-1)}}} Divide both sides by {{{-1}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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{{{35x-30y=65}}} Now go back to the first equation.



{{{35(1)-30y=65}}} Plug in {{{x=1}}}.



{{{35-30y=65}}} Multiply.



{{{-30y=65-35}}} Subtract {{{35}}} from both sides.



{{{-30y=30}}} Combine like terms on the right side.



{{{y=(30)/(-30)}}} Divide both sides by {{{-30}}} to isolate {{{y}}}.



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



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



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



{{{drawing(500,500,-9,11,-11,9,
grid(1),
graph(500,500,-9,11,-11,9,(13-7x)/(-6),(11-6x)/(-5)),
circle(1,-1,0.05),
circle(1,-1,0.08),
circle(1,-1,0.10)
)}}} Graph of {{{7x-6y=13}}} (red) and {{{6x-5y=11}}} (green)