Variable Denominators

Let us assume that $\frac{1}{x}=m$ and $\frac{1}{y}=n$ . Therefore, it is evident that $2m-24n=-1$ and $6m+33n=2$ . Furthermore, we get the system

$\left\{ \begin{array}{l l} 2m-24n=-1 & \quad \\ 6m+33n=2 & \quad \end{array} \right.$

whose solutions are $m=\frac{1}{14}$ and $n=\frac{1}{21}$ , meaning that $x=14$ and $y=21$ . Thus, the product $xy=294$ .

Let $\frac{1}{x} = a$ and $\frac{1}{y} = b$ , then:

$2a - 24b = -1$ ...( i )

$6a + 33b = 2$ ...( ii )

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Eliminate ( i ) with ( ii )

$2a - 24b = -1$ | $\times 3$

$6a + 33b = 2$ | $\times 1$

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$6a - 72b = -3$

$6a + 33b = 2$

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$-105b = -5$

$b = \frac{1}{21}$ ( remember $b = \frac{1}{y}$ )

$\frac{1}{y} = \frac{1}{21}$

$y = 21$

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Substitute the value of $b$ to ( i )

$2a - 24b = -1$

$2a - ( 24 \times \frac{1}{21} ) = -1$

$2a - \frac{8}{7} = -1$

$2a = -1 + \frac{8}{7}$

$2a = \frac{1}{7}$

$a = \frac{1}{14}$ ( remember $a = \frac{1}{x}$ )

$\frac{1}{x} = \frac{1}{14}$

$x = 14$

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So, $xy = 14 \times 21 = 294$

Multiply equation A by 3, so that both equations have the term 6/x.

Next, take both equations and subtract them from each other. Order doesn't matter. Then, solve for y. Once you have y, solve for x. It is 14 and 21.

Let's work with the first equation first,

$\frac {2}{x} - \frac {24}{y} = -1 \Longrightarrow \frac {2y-24x}{xy} = -1 \Longrightarrow 2y-24x = -xy$

$\Longrightarrow 2y+xy=24x \Longrightarrow y(2+x)=24x \Longrightarrow y = \frac {24x}{2+x}$

Now, let's substitute the value of $y$ to the second equation,

$\frac {6}{x} + \frac {33}{y} =2 \Longrightarrow \frac {6}{x} + \frac {33}{\frac {24x}{2+x}} = 2\Longrightarrow \frac {6}{x} + \frac {66+33x}{24x} = 2 \Longrightarrow \frac {144+66+33x}{24x} = 2$

$\Longrightarrow 210 + 33x = 48x \Longrightarrow 15x = 210 \Longrightarrow x = 14$

Next, let's again substitute the value of $x$ in the equation $y = \frac {24x}{2+x}$ ,

$y = \frac {24x}{2+x} \Longrightarrow y = \frac {24 \times 14}{2+14} \Longrightarrow y= \frac {336}{16} \Longrightarrow y = 21$

Hence, the product $xy$ is, $14 \times 21 = 294$

Therefore, the required answer is $\fbox {294}$ ...

2/x=24/y-1 3.(2/x)+33/y=2 logo 3.(24/y-1)+33/y=2 então y=21 Como 2/x=24/21-1 implica que x=14 O produto x.y é 21 . 14 = 294

given 2/x-24/y=-1 & 6/x+33/y=2

2y-24x=-xy & 6y+33x=2xy

using simultaneous linear equations , we get:

6y-72x=-3xy - equation 1

6y+33x=2xy - equation 2

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-105x=-5xy

=> y=21

substitute y in equation 2

6y+33x=2xy

=>126+33x=42x

=>9x=126

=>x=14

=>xy=21*14=294

required answer is 294

2y-24x=-xy (*-3) 6y+33x=2xy

72x+33x=5xy 105x=5xy y=21 x=2y/(24-y) => x=14 x*y = 294

Let us multiply by xy. As a result, we get the equations $2y-24x=-xy$ and $6y+33x=2xy$ . Adding these up, we have $xy=9x+8y$ . Let's substitute this in for xy. So

$24x-2y=8y+9x$

$15x=10y$

$x=\frac{2}{3}y$

Substituting this back in, we get

$16y-2y=xy$

$y(14-x)=0$

$x=14$ $y=21$ .

From equation 1, xy = 24x - 2y (3) From equation 2, 2xy = 33x + 6y (4)

From equation 3 and 4, x = 2/3*y

then from equation 3, y = 21 So, x = 2/3 * 21 = 14

Therefore, xy = 14 * 21 = 294

first make a straight simple equations as 2y-24x=-xy and 6y+33x=2xy then solve them to find either x or y. And substitute the value of a n unknown variable in either of the equation to find the other.Atlast find the product of x*y here x=42/3 and y=21,so xy=294 is the answer.

2y - 24x = - xy......(eq 1)

6y + 33x = 2xy... (eq 2)

6y + 33x = 2(-2y + 24x)

6y + 33x = -4y + 48x

10y = 15x

x = (10/15) y

x = (2/3)y

and then subs x to eq (1),so :

2y - 24 (2/3)y = -(2/3)y . y

2y - 16y = - (2/3) y^2

-14y = - (2/3) y^2

14(3/2) = y

y = 21

so that x = (2/3) 21 = 14

the result is xy = (14) (21) = 294

I used elimination. 2/x - 24/y = -1 6/x + 33/y = 2 You have to equate the numerator numbers. So, I mutiply 2/x - 24/y = -1 with 3. It became 6x - 72/y = -3, it's similar with 6/x + 33/y = 2. (used X for elimination).

evaluate (6/x - 72/y = -3) - (6x + 33/y = 2) . The result is y=21. distribute coefficient of y or 21 to 6/x + 33/y = 2 became 6/x + 33/21 = 2 you must transfer 33/21 to right, 6/x = 2 - 33/21. You have to equate the denominator number. 6/x = 2-3321 6/x = (42-33)/21 6/x = 9/21 x = 14 Finally, x = 14 and y = 21, so xy =294

BY SOLVING THE GIVEN EQUATIONS WE FIND X AND Y WHICH AR X=14 AND Y=21 AND THEN XY=294

$2/x - 24/y = -1$ ....... i

$6/x + 33/y = 2$ ........ii

$i * 3$ => $6/x - 72/y = -3$ ....a

$ii * 1$ => $6/x + 33/y = 2$ .....b

$a - b$ => $-72/y - 33/y = -3 -2$ => $-105/y = -5$ => $105 = 5y$ => $21 = y$

Putting the value of y in the 1st equation => $2/x - 24/21 = -1$ => $2/x = 8/7 - 1$ => $2/x = 1/7$ => $14 = x$

And xy = $21 * 14 = 294$

make a system you will found 6/x= 72/y -3 after this 72/y-3 + 33/y = 2 y =21

2/x- 24/22= -1 x=14

Please note that this solution only works because the variables were non-zero. All sorts of strange results can be produced by dividing by zero.

1. Clear the fractions by multiplying both equations by xy . This yields the following system of equations

   2y - 24x = -xy

   6y + 33x = 2xy

2. Solve either equation for either variable. I solved the first equation for y.

   y = 12x - $\frac{xy}{2}$

3. Substitute this value into the other equation.

   6(12x - $\frac{xy}{2}$ ) + 33x = 2xy

4. Simplify

   72x - 3xy + 33x = 2xy

   105x -3xy = 2xy

   105x = 5xy

   105 = 5y (This is where you have to know that x is non-zero)

   21 = y

5. Substitute this value for y into any of the above equations and solve for x.

   21 = 12x - $\frac{21x}{2}$

   42 = 24x - 21x

   42 = 3x

   14 = x

6. Now find the product of x and y and you're done.

Multiplicando ambas as equações por \­( xy \­), temos: \­( 2 y - 24 x = - xy\­) e \­( 6 y + 33 y = 2xy (I)\­). Multiplicando a primeira por \­( -3 \­), temos \­( -6 y + 72 x = + 3xy (II) \­). Somando \­( (I) \­) e \­( (II) \­), chegamos em $105 x = 5 xy$ de onde chegamos em $y=21$ . Substituindo na primeira equação, chegamos em \­( x = 14.\­). Portanto, \­( x \times y = 294\­).