Question 1180819


Find the constant m for which all three lines 

{{{x - y = -m}}} …eq. 1 
{{{ 2x + y = m - 1 }}}…eq. 2
{{{x + 5y = 4m + 1}}} …eq. 3 
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{{{x - y = -m}}} …eq. 1 , solve for {{{x}}}

{{{x  = y-m}}} …eq. 1a 

{{{ 2x + y = m - 1 }}}…eq. 2, substitute  {{{x}}}

{{{ 2(y-m) + y = m - 1 }}}

{{{ 2y-2m + y = m - 1 }}}

{{{ 3y = 2m+m - 1 }}}

{{{ 3y = 3m - 1 }}}

{{{ y = m - 1/3 }}}........eq,2a

go to

{{{x  = y-m}}} …eq. 1a , substitute  {{{y}}}

{{{x  = m - 1/3 -m}}} 

{{{x  = - 1/3 }}} ..................eq.1b

go to

{{{x + 5y = 4m + 1}}} …eq. 3 , substitute {{{x}}} and {{{y}}}

{{{-1/3 + 5(m - 1/3) = 4m + 1}}}...solve for {{{m}}}


{{{-1/3 + 5m - 5/3 = 4m + 1}}}

{{{ 5m - 2 = 4m + 1}}}

{{{ 5m - 4m = 2 + 1}}}

{{{ m = 3}}}

go to
{{{ y = m - 1/3 }}}........eq,2a , substitute {{{m}}}

{{{ y = 3 - 1/3 }}}

{{{ y = 8/3 }}}


{{{x - y = -3}}} …eq. 1 
{{{ 2x + y = 3 - 1 }}}…eq. 2
{{{x + 5y = 4*3 + 1}}} …eq. 3 
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{{{x - y = -3}}} …eq. 1 
{{{ 2x + y = 2 }}}…eq. 2
{{{x + 5y = 13}}} …eq. 3 
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all three lines are intersecting at one point which is ({{{-1/3}}}, {{{ 8/3}}})


{{{ drawing( 600, 600, -10, 10, -10, 10,
circle(-1/3,8/3,.12), locate(-1/3,8/3,p(-1/3,8/3)),

graph( 600, 600, -10, 10, -10, 10, x+3, -2x+2,-x/5+13/5)) }}}