Question 824664
First find the intersection:
Add the equations:
{{{ 5x - 3y + 2 = 0 }}} 
{{{ x + 3y + 1 = 0 }}}
{{{ 6x +3 = 0 }}}
{{{ 6x = 3 }}}
{{{ x = 1/2 }}}
and
{{{ x + 3y + 1 = 0 }}}
{{{ 1/2 + 3y + 1 = 0 }}}
{{{ 3y = -3/2 }}}
{{{ y = -1/2 }}}
The intersection is (1/2, -1/2 )
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Now find the slope of {{{ 2x + 3y + 3 = 0 }}}
{{{ 3y = -2x - 3 }}}
{{{ y = -(2/3)*x - 1 }}}
The slope is {{{ m = -2/3 }}}
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I can write the general point-slope equation:
{{{ ( y - (-1/2) ) / ( x - 1/2 ) = -2/3 }}}
{{{ y + 1/2 = -2/3*( x - 1/2 ) }}}
{{{ y + 1/2 = (-2/3)*x + 1/3 }}}
Multiply both sides by {{{ 6 }}}
{{{ 6y + 3 = -4x + 2 }}}
{{{ 6y = -4x -1 }}}
{{{ y = (-2/3)*x - 1/6 }}} answer
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check:
does it go through (1/2, -1/2) ?
{{{ y = (-2/3)*x - 1/6 }}} 
{{{ -1/2 = (-2/3)*(1/2) - 1/6 }}} 
{{{ -1/2 = -1/3 - 1/6 }}}
{{{ -3/6 = -2/6 - 1/6 }}}
{{{ -3/6 = -3/6 }}}
OK