Question 1175505
Given the linear system

{{{x + 2y = 10}}}
{{{3x + (6 + t)y = 30}}}
-----------------------------

(a) Determine a particular value of t so that the system
has infinitely many solutions.

Lines that lay right on top of each other; the linear system has infinitely many solutions, basically it's same line 

first equation have {{{x}}} and second equation have {{{3x}}}, first equation have constant term {{{10}}} and second equation have {{{30}}}=> so, multiplied by {{{3}}}

then must be {{{2*3=6+t}}}->{{{6=6+t}}}->{{{t=0}}}

{{{x + 2y = 10}}}..........{{{t=0}}}
{{{3x + (6 + 0)y = 30}}}
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{{{x + 2y = 10}}}
{{{3x + 6y = 30}}}
------------------------=>the linear system has {{{infinitely}}} many solutions

{{{ graph( 600,600, -10, 10, -10, 10, -x/2+5, -3x/6+30/6) }}}


(b) Determine a particular value oft so that the system
has a unique solution.

{{{t}}} could any number except {{{0}}}
{{{x + 2y = 10}}}..........let's take {{{t=2}}}
{{{3x + (6 +2)y = 30}}}
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{{{x + 2y = 10 }}}
{{{3x + 8y = 30}}}
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{{{ graph( 600,600, -10, 20, -10, 20, -x/2+5, -3x/8+30/8) }}}


(c) How many different values of {{{t}}} can be selected in
part (b)?
   
{{{infinitely}}} number of different values of {{{t }}}can be selected