Solve: 6x + 3y = 7xy and 3x + 9y =11xy - Vidyadip

Question

Solve: 6x + 3y = 7xy and 3x + 9y =11xy

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Answer

The given equations are as follow:
6x + 3y = 7xy ...(i)
3x + 9y = 11 ....(ii)
For equation (i), we have:
$\frac{6\text{x}+3\text{y}}{\text{xy}}=7$
$\frac{6\text{x}}{\text{xy}}+\frac{3\text{y}}{\text{xy}}=7$
$\frac{6\text{}}{\text{y}}+\frac{3\text{}}{\text{x}}=7...(\text{iii})$
For equation (ii), we have:
$\frac{3\text{x}+9\text{y}}{\text{xy}}=11$
$\frac{3\text{x}}{\text{xy}}+\frac{9\text{y}}{\text{xy}}=11$
$\frac{3\text{}}{\text{y}}+\frac{9\text{}}{\text{x}}=11...(\text{iv})$
On Substituting $\frac{1}{\text{y}}=\text{v}$ and $\frac{1}{\text{x}}=\text{u}$ in (iii) and (iv), we get:
6v + 3u = 7 ...(v)
3v + 9u = 11 ....(vi)
On multipiying (v) by 3, We get:
18v + 9u = 21 ....(vii)
On subtracting (vi) from (vii), we get:
15v = 10
$\Rightarrow\text{v}=\frac{10}{15}=\frac{2}{3}$
$\Rightarrow\frac{1}{\text{y}}=\frac{2}{3}$
$\Rightarrow\text{y}=\frac{3}{2}$
On subtracting $\text{y}=\frac{3}{2}$ in (iii), we get:
$\frac{6}{\frac{3}{2}}+\frac{3}{\text{x}}=7$
$\Rightarrow4+\frac{3}{\text{x}}=7$
$\Rightarrow\frac{3}{\text{x}}=3$
$\Rightarrow3\text{x}=3$
$\Rightarrow\text{x}=1$
Hence, the required solution is x = 1 and $\text{y}=\frac{3}{2}$

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