Question

If 5x + 7y = 6xy & 4y - 3x = 3xy then solution of this system of equations is

Answer 1

Given:-

$• \: \: \: \: \: \: \: 5x + 7y = 6xy$

$\implies \: \frac{5x + 7y}{xy} = 6$

$\implies \: \frac{5x}{xy} + \frac{7y}{xy} = 6$

$\implies \: \color{red} \bold { \frac{5}{y} + \frac{7}{x} = 6} \: \: - - - - - (1)$

And

$• \: \: \: \: 4y - 3x = 3xy$

$\implies \frac{4x - 3y}{xy} = 3$

$\implies \: \frac{4x}{xy} - \frac{3y}{xy} = 3$

$\implies \: \: \color{red} \bold{\frac{4}{y} - \frac{3}{x} = 3} \: \: - - - - (2)$

Suppose ,

$\bold{‡ \: \: \: \: \frac{1}{y} \rightarrow \: u \: }$

$\bold{‡ \: \: \: \: \frac{1}{x} \rightarrow \: v \: }$

So,

• eq(1) → 5u + 7v = 6 ----------(3)

• eq(2) → 4u - 3v = 3. ------------(4)

{ we are going to solve those equations by substracting method }

Now,

eq(3) × 4 → 20u + 28v = 24 ---------(5)

eq(4) × 5 → 20u - 15v = 15 -----------(6)

~ eq(5) - eq(6) ↓

→ 28v + 15 v = 9

→ 43 v = 9

→ v = 9/43

By Substituting the value of v in eq (3) we get,

$5u + \frac{7 \times 9}{43} = 6$

$\implies \: 5u + \frac{63}{43} = 6$

$\implies \: 5u = 6 - \frac{63}{43}$

$\implies \: 5u = \frac{195}{43}$

$\implies \: \underline \bold{u \: = \frac{39}{43} }$

Since ,

~ u = 1/y

→ y = 1/u

→ y = 43/39

And

v = 1/x

→ x = 1/v

→ x = 43/9