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Find the two values of y which satisfy the simultaneous equa...

Find the two values of y which satisfy the simultaneous equation 3x + y = 8, x\(^2\) + xy = 6.

A.

-1 and 5

B.

-5 and 1

C.

1 and 5

D.

1 and 1

Correct answer is A

\(3x + y = 8 ... (i)\)

\(x^{2} + xy = 6 ... (ii)\)

From (i), \(y = 8 - 3x\)

From (ii), \(xy = 6 - x^{2} \implies y = \frac{6 - x^{2}}{x}\)

Equating the two values of y, we have

\(8 - 3x = \frac{6 - x^{2}}{x} \implies x(8 - 3x) = 6 - x^{2}\)

\(8x - 3x^{2} = 6 - x^{2} \implies 6 - x^{2} - 8x + 3x^{2} = 0\)

\(2x^{2} - 8x + 6 = 0\)

\(x^{2} - 4x + 3 = 0\)

\(x^{2} - 3x - x + 3 = 0 \implies x(x - 3) - 1(x - 3) = 0\)

\((x - 1)(x - 3) = 0 \therefore \text{x = 1 or 3}\)

\(y = 8 - 3x \)

When x = 1, \(y = 8 - 3(1) = 5\)

When x = 3, \(y = 8 - 3(3) = -1\)

\(\therefore \text{y = -1 or 5}\)