(a) Given that \(\frac{5y - x}{8y + 3x} = \frac{1}{5}\), find the value of \(\frac{x}{y}\) to two decimal places.
(b) If 3 is a root of the quadratic equation \(x^{2} + bx - 15 = 0\), determine the value of b. Find the other root.
(a) \(\frac{5y - x}{8y + 3x} = \frac{1}{5}\)
\(5(5y - x) = 8y + 3x \implies 25y - 5x = 8y + 3x\)
\(25y - 8y = 3x + 5x \implies 17y = 8x\)
\(\therefore \frac{x}{y} = \frac{17}{8} = 2.125 \approxeq 2.13\)
(b) \(x^{2} + bx - 15 = 0\)
Since x = 3 is a root of the equation, f(3) = 0.
\(3^{2} + 3b - 15 = 0 \implies 3b = 6\)
\(b = 2\)
\(\therefore\) The equation is \(x^{2} + 2x - 15 = 0\)
\(x^{2} - 3x + 5x - 15 = 0 \implies x(x - 3) + 5(x - 3) = 0\)
\((x - 3)(x + 5) = 0\)
The second root of the equation is x = -5.
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