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Question

Solve the following system of equations:
8v=3u+5uv
6v5u=2uv.

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Solution

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

If we put u=0 in either of the two equations, we get v=0

Thus, u=0,v=0 from one solution of the given system of equations.

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

Since u0, v0. Therefore, uv0

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

8u3v=5(i)

6u5v=2(ii)

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

8x3y=5(iii)
6x5y=2(iv)

Multiplying equation (i) by 3 and equation (ii) by 4, we get

24x9y=15(v)

24x20y=8(vi)

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

11y=23y=2311

Putting y=2311 in equation (iii), we get

8x6911=5

8x=6911+5

8x=12411

x=3122

Now,
x=3122

1u=3122

u=2231

And,
y=2311

1v=2311

v=1123

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

(i) u=0,v=0

(ii) u=2231,v=1123.

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