**Find the value of x and y in the simultaneous equation: 3x + y = 21; xy = 30**

### Question

Find the value of x and y in the simultaneous equation: 3x + y = 21; xy = 30

### Options

A) x = 3 or 7, y = 12 or 8

B) x = 6 or 1, y = 11 or 5

C) x = 2 or 5, y = 15 or 6

D) x = 1 or 5, y = 10 or 7

The correct answer is C.

### Explanation:

3x + y = 21 ... (i);

xy = 30 ... (ii)

From (ii), \(y = \frac{30}{x}\). Putting the value of y in (i), we have

3x + \(\frac{30}{x}\) = 21

\(\implies\) 3x\(^2\) + 30 = 21x

3x\(^2\) - 21x + 30 = 0

3x\(^2\) - 15x - 6x + 30 = 0

3x(x - 5) - 6(x - 5) = 0

(3x - 6)(x - 5) = 0

3x - 6 = 0 \(\implies\) x = 2.

x - 5 = 0 \(\implies\) x = 5.

If x = 2, y = \(\frac{30}{2}\) = 15;

If x = 5, y = \(\frac{30}{5}\) = 6.

xy = 30 ... (ii)

From (ii), \(y = \frac{30}{x}\). Putting the value of y in (i), we have

3x + \(\frac{30}{x}\) = 21

\(\implies\) 3x\(^2\) + 30 = 21x

3x\(^2\) - 21x + 30 = 0

3x\(^2\) - 15x - 6x + 30 = 0

3x(x - 5) - 6(x - 5) = 0

(3x - 6)(x - 5) = 0

3x - 6 = 0 \(\implies\) x = 2.

x - 5 = 0 \(\implies\) x = 5.

If x = 2, y = \(\frac{30}{2}\) = 15;

If x = 5, y = \(\frac{30}{5}\) = 6.

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3x + y = 21 ... (i);

xy = 30 ... (ii)

From (ii), \(y = \frac{30}{x}\). Putting the value of y in (i), we have

3x + \(\frac{30}{x}\) = 21

\(\implies\) 3x\(^2\) + 30 = 21x

3x\(^2\) - 21x + 30 = 0

3x\(^2\) - 15x - 6x + 30 = 0

3x(x - 5) - 6(x - 5) = 0

(3x - 6)(x - 5) = 0

3x - 6 = 0 \(\implies\) x = 2.

x - 5 = 0 \(\implies\) x = 5.

If x = 2, y = \(\frac{30}{2}\) = 15;

If x = 5, y = \(\frac{30}{5}\) = 6.