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Math Help - Grade 9 Equations with Fractions

  1. #1
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    Grade 9 Equations with Fractions

    Hi all.
    I have an ext. exam coming up tomorrow and I needed help with Equations with Fractions.
    I looked up a tutorial and got this, which I do not understand:
    x
    3
    + x − 2
    5
    = 6
    Multiply both sides of the equation -- every term -- by the LCMof denominators. Every denominator will then cancel. We will then have an equation without fractions.

    The LCM of 3 and 5 is 15. Therefore, multiply every term on both sides by 15:
    15 x
    3
    + 15 x − 2
    5
    = 15 6
    Each denominator will now cancel into 15 -- that is the point -- and we have the following simple equation that has been "cleared" of fractions:
    5x + 3(x − 2) = 90.
    How do they multiply the terms by 15?
    Like how do they multiply 15 by x over 3?
    This is confusing. :P

    Thanks in advance!
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  2. #2
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    Re: Grade 9 Equations with Fractions

    Quote Originally Posted by JesseElFantasma View Post
    Like how do they multiply 15 by x over 3?
    I don't understand your question. What do you mean by "how"? One is allowed to multiply any number by any other number; multiplication is defined for every two numbers. Do you mean "How to simplify 15\frac{x}{3}"? Do you mean "Why 15\frac{x}{3}=5x"? Do you mean "Why is \frac{x}{3}+\frac{x-2}{5}=6 equivalent to 15\frac{x}{3}+15\frac{x-2}{5}=15\cdot6"? What exactly is your question?
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  3. #3
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    Re: Grade 9 Equations with Fractions

    Quote Originally Posted by emakarov View Post
    I don't understand your question. What do you mean by "how"? One is allowed to multiply any number by any other number; multiplication is defined for every two numbers. Do you mean "How to simplify 15\frac{x}{3}"? Do you mean "Why 15\frac{x}{3}=5x"? Do you mean "Why is \frac{x}{3}+\frac{x-2}{5}=6 equivalent to 15\frac{x}{3}+15\frac{x-2}{5}=15\cdot6"? What exactly is your question?
    Yes, why does 15 multiplied by x over 3 equal 5x ?
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  4. #4
    MHF Contributor Reckoner's Avatar
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    Re: Grade 9 Equations with Fractions

    Quote Originally Posted by JesseElFantasma View Post
    Yes, why does 15 multiplied by x over 3 equal 5x ?
    Do you know how to divide 15 by 3?

    15\cdot\frac x3

    =\frac{15}1\cdot\frac x3

    =\frac{15\cdot x}{1\cdot3}

    =\frac{15x}3

    =\frac{5x}1 = 5x
    Thanks from JesseElFantasma
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  5. #5
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    Re: Grade 9 Equations with Fractions

    It may be more convenient to break division into two operations: multiplication and reciprocal function. Thus, x / y would really mean x * (1 / y). This is, in fact, how it is done in higher mathematics.

    As you know, multiplication obeys the following laws:

    x * y = y * x (commutativity)
    x * (y * z) = (x * y) * z (associativity)

    Therefore,

    15\frac{x}{3} = 15 * (x / 3) =
    15 * (x * (1 / 3)) = (by definition of division)
    15 * ((1 / 3) * x) = (by commutativity)
    (15 * (1 / 3)) * x = (by associativity)
    5 * x

    because 15 * (1 / 3) = 15 / 3 = 5.
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  6. #6
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    Re: Grade 9 Equations with Fractions

    Think of x as a brick, (or some other object if you prefer it).

    x divided by 3 means that the brick is being divided into three pieces, \frac{x}{3} is a third of a brick.

    Now suppose that you have 15 of these and that they could be stuck back together again. How many whole bricks could you make ?

    Answer 5 bricks. Algebraically, 5x.

    For the next term, relate (x-2) to some object.
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