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Math Help - solve the equation floor

  1. #1
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    solve the equation floor

    solve the equation

    (19x+16)/10=[(4x+7)/3]
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  2. #2
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    (19x + 16)/10 - (4x+7)/3 = 0

    (57x + 48)/30 - (40x+70)/30 = 0

    ((17x - 22)/30)*30 = 0*30

    17x - 22 = 0

    (17x)/17 = 22/17

    x = 22/17
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  3. #3
    Senior Member JaneBennet's Avatar
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    millerst, the RHS is \left\lfloor\frac{4x+7}3\right\rfloor, not \frac{4x+7}3.

    Quote Originally Posted by perash View Post
    solve the equation

    (19x+16)/10=[(4x+7)/3]
    Let \left\lfloor\frac{4x+7}3\right\rfloor=n.

    Then n\ \leqslant\ \frac{4x+7}3\ <\ n+1

    \Rightarrow\ 3n\ \leqslant\ 4x+7\ <\ 3n+3

    \Rightarrow\ \frac{3n-7}4\ \leqslant\ x\ <\ \frac{3n-4}4

    \Rightarrow\ \frac{57n-69}4\ \leqslant\ 19x+16\ <\ \frac{57n-12}4

    \Rightarrow\ \frac{57n-69}{40}\ \leqslant\ \frac{19x+16}{10}\ <\ \frac{57n-12}{40}

    \Rightarrow\ \frac{57n-69}{40}\ \leqslant\ n\ <\ \frac{57n-12}{40}

    \Rightarrow\ 57n-69\ \leqslant\ 40n\ <\ 57n-12

    \Rightarrow\ -69\ \leqslant\ -17n\ <\ -12

    \Rightarrow\ \frac{12}{17}\ <\ n\ \leqslant\ \frac{69}{17}

    Hence n=1,2,3,4. The four solutions are found by solving \frac{19x+16}{10}=1,2,3,4 for x.
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  4. #4
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    Quote Originally Posted by perash View Post
    solve the equation

    (19x+16)/10=[(4x+7)/3]
    I take it from the title that this is really supposed to be
    (19x+ 16)/10= \lfloor (4x+7)/3\rfloor.

    The first thing that tells us is that right hand side is an integer and so (19x+ 16)/10 must be an integer: (19x+ 16)/10= n so 19x+ 16= 10n for some integer n. From that 19x= 10n-16 and x= (10n- 16)/19.

    Putting that into the right side, (4x+ 7)/3= (40n+ 69)/57 and that must have "floor" n. That is, it must be equal to n+ \delta where \delta is between 0 and 1.

    (40n+ 69)/57= n+ \delta so 40n+ 69= 57n+ 5\delta or 17n= 69- 57\delta and, finally, n= \frac[99- 57\delta}{17}.

    Since the largest \delta can be is 1, n cannot be smaller than (69-57)/17= 12/17= 0.705 .... Since the smallest \delta can be is 0, n cannot be larger than 69/17= 4.04. Since n must be a constant n must be 1, 2, 3, or 4.

    If n= 1, then 19x+ 16= 10 and x= -6/19.

    If n= 2, then 19x+ 16= 20 and x= 4/19.

    If n= 3, then 19x+ 16= 30 and x= 14/19.

    If n= 4, then 19x+ 16= 40 and x= 24/19.

    I will leave it to you to check that all those do, in fact, satisfy the original equation.

    Blast, too slow!! Ah, well, ladies first.
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