Question 198232


Start with the given system of equations:

{{{system(5x-6y=-123,3x+13y=-107)}}}



{{{3(5x-6y)=3(-123)}}} Multiply the both sides of the first equation by 3.



{{{15x-18y=-369}}} Distribute and multiply.



{{{-5(3x+13y)=-5(-107)}}} Multiply the both sides of the second equation by -5.



{{{-15x-65y=535}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x-18y=-369,-15x-65y=535)}}}



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



{{{(15x-18y)+(-15x-65y)=(-369)+(535)}}}



{{{(15x+-15x)+(-18y+-65y)=-369+535}}} Group like terms.



{{{0x+-83y=166}}} Combine like terms.



{{{-83y=166}}} Simplify.



{{{y=(166)/(-83)}}} Divide both sides by {{{-83}}} to isolate {{{y}}}.



{{{y=-2}}} Reduce.



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{{{15x-18y=-369}}} Now go back to the first equation.



{{{15x-18(-2)=-369}}} Plug in {{{y=-2}}}.



{{{15x+36=-369}}} Multiply.



{{{15x=-369-36}}} Subtract {{{36}}} from both sides.



{{{15x=-405}}} Combine like terms on the right side.



{{{x=(-405)/(15)}}} Divide both sides by {{{15}}} to isolate {{{x}}}.



{{{x=-27}}} Reduce.



So the solutions are {{{x=-27}}} and {{{y=-2}}}.



Which form the ordered pair *[Tex \LARGE \left(-27,-2\right)].



This means that the system is consistent and independent.