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Math Help - Algebra - Linear systems

  1. #1
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    Algebra - Linear systems

    1.
    a) For what value of a is there no unique solution for the following simultaneous equations.
    3x - 2y = 5
    5x + ay = 4

    b) When a has this value is there an infinite number of solutions or no solution? Justify your answer.

    2.
    Solve for x: 3 - x/4 < 2 - 3x/5
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  2. #2
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    Quote Originally Posted by LilDragonfly
    1.
    a) For what value of a is there no unique solution for the following simultaneous equations.
    3x - 2y = 5
    5x + ay = 4

    b) When a has this value is there an infinite number of solutions or no solution? Justify your answer.

    2.
    Solve for x: 3 - x/4 < 2 - 3x/5
    1.
    That happens when the determinant is zero, thus,
    \left| \begin{array}{cc}3&-2\\5&a \end{array} \right|=0
    Thus,
    3a+10=0
    Thus,
    a=-\frac{10}{3}.

    To determine if this system is consistent or inconsistent substitute the value of a yields,
    \left\{ \begin{array}{c} 3x-2y=5\\15x-10y=12

    Note:I multiplied to second equation by 3 to clear denominator.

    Now, multiply the first equation by 5 to yield,
    \left\{ \begin{array}{c} 15x-10y=25\\15x-10y=12

    How can 15x-10y=25\mbox{ and }12?
    Impossible, thus this system is inconsistent.
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  3. #3
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    2.
    You have,
    3-x/4<2-3x/5
    Multiply by LCD=20,
    60-5x<40-12x
    Add 5x to both sides,
    60<40-7x
    Subtract 40 from both sides,
    20<-7x
    Divide both sides by -7 remember to flip sign ,
    -20/7>x
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