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Question

Solve:
3(2u+v)=7uv and
3(u+3v)=11uv.

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Solution

Clearly, the given equations are not linear equations in the variables u and v but can be reduced to linear equations by appropriate substitution.

If we put u=0 in either of the two equations, we get v=0.
So, u=0,v=0 form a solution of the given system of equations.

To find the other solutions, we assume that u0,v0.
Now, u0,v0uv0.

On dividing each one of the given equations by uv, we get

6v+3u=7 (i)

3v+9u=11 (ii)

Taking 1u=x and 1v=y, the given equations become

3x+6y=7 ..(iii)

9x+3y=11 .(iv)

Multiplying equation (iv) by 2, the given system of equations becomes

3x+6y=7 .(v)

18x+6y=22 .(vi)

Substracting equation (vi) from equation (v), we get

15x=15x=1

Putting x=1 in equation (iii), we get

3+6y=7y=46=23

Now, x=11u=1u=1

and, y=231v=23v=32.

Hence, the given system of equations has two solutions given by

(i) u=0,v=0

(ii) u=1,v=3/2.

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