Question 82846
Start with the given system of equations

{{{x/3 + y/2 = 1/6}}}
{{{x - 6y = 8 }}}


Multiply the top equation by 6 to get integer coefficients (this eliminates the denominator)

{{{6(x/3 + y/2 )= 6(1/6)}}} Multiply both sides by 6

{{{2x+3y= 1}}} Distribute


So now the system becomes 



{{{2x+3y= 1}}}
{{{x - 6y = 8 }}}


*[invoke solving_linear_system_by_elimination 2, 3, 1, 1, -6, 8]

Check:

Plug in (2,-1) into the system

{{{x/3 + y/2 = 1/6}}}
{{{x - 6y = 8 }}}

{{{2/3 + -1/2 = 1/6}}} Plug in x=2, y=-1
{{{2 - 6(-1) = 8 }}}

{{{4/6 - 3/6 = 1/6}}}
{{{2 + 6 = 8 }}}

{{{1/6 = 1/6}}} works
{{{8= 8 }}} works

this verifies our answer


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Now lets look at


{{{3x + 5y = 12}}}
{{{7x + 5y = 8}}}


*[invoke solving_linear_system_by_elimination 3, 5, 12, 7, 5, 8]

Check:

Plug in (-1,3) into the system

{{{3x + 5y = 12}}}
{{{7x + 5y = 8}}}

{{{3(-1) + 5(3) = 12}}} Plug in x=2, y=-1
{{{7(-1) + 5(3) = 8}}}

{{{-3 +15 = 12}}}
{{{-7 +15 = 8 }}}

{{{12 = 12}}} works
{{{8= 8 }}} works

this verifies our answer