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:

Quote:

**Multiply both sides of the equation -- every term -- by the LCM***of 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 pointhttp://www.themathpage.com/acalc/calc_IMG/excl.gif -- and we have the following simple equation that has been "cleared" of fractions:

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! :)

Re: Grade 9 Equations with Fractions

Quote:

Originally Posted by

**JesseElFantasma** 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 $\displaystyle 15\frac{x}{3}$"? Do you mean "Why $\displaystyle 15\frac{x}{3}=5x$"? Do you mean "Why is $\displaystyle \frac{x}{3}+\frac{x-2}{5}=6$ equivalent to $\displaystyle 15\frac{x}{3}+15\frac{x-2}{5}=15\cdot6$"? What exactly is your question?

Re: Grade 9 Equations with Fractions

Quote:

Originally Posted by

**emakarov** 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 $\displaystyle 15\frac{x}{3}$"? Do you mean "Why $\displaystyle 15\frac{x}{3}=5x$"? Do you mean "Why is $\displaystyle \frac{x}{3}+\frac{x-2}{5}=6$ equivalent to $\displaystyle 15\frac{x}{3}+15\frac{x-2}{5}=15\cdot6$"? What exactly is your question?

Yes, why does 15 multiplied by x over 3 equal 5x ?

:)

Re: Grade 9 Equations with Fractions

Quote:

Originally Posted by

**JesseElFantasma** Yes, why does 15 multiplied by x over 3 equal 5x ?

:)

Do you know how to divide 15 by 3?

$\displaystyle 15\cdot\frac x3$

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

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

$\displaystyle =\frac{15x}3$

$\displaystyle =\frac{5x}1 = 5x$

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,

$\displaystyle 15\frac{x}{3}$ = 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.

Re: Grade 9 Equations with Fractions

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

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

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

Answer $\displaystyle 5$ bricks. Algebraically, $\displaystyle 5x.$

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