1. **State the problem:** Solve the simultaneous equations for real numbers $p, q \neq 0$: $$4p + q = 10$$ $$\frac{3}{p} - \frac{4}{q} = 1$$ 2. **Express $q$ from the first equation:** $$q = 10 - 4p$$ 3. **Substitute $q$ into the second equation:** $$\frac{3}{p} - \frac{4}{10 - 4p} = 1$$ 4. **Multiply both sides by $p(10 - 4p)$ to clear denominators:** $$3(10 - 4p) - 4p = p(10 - 4p)$$ 5. **Expand both sides:** $$30 - 12p - 4p = 10p - 4p^2$$ Simplify left side: $$30 - 16p = 10p - 4p^2$$ 6. **Bring all terms to one side:** $$30 - 16p - 10p + 4p^2 = 0$$ $$4p^2 - 26p + 30 = 0$$ 7. **Divide entire equation by 2 to simplify:** $$\cancel{2} \times 2p^2 - \cancel{2} \times 13p + \cancel{2} \times 15 = 0 \Rightarrow 2p^2 - 13p + 15 = 0$$ 8. **Solve quadratic equation $2p^2 - 13p + 15 = 0$ using the quadratic formula:** $$p = \frac{13 \pm \sqrt{(-13)^2 - 4 \times 2 \times 15}}{2 \times 2} = \frac{13 \pm \sqrt{169 - 120}}{4} = \frac{13 \pm \sqrt{49}}{4}$$ $$p = \frac{13 \pm 7}{4}$$ 9. **Calculate the two possible values for $p$:** - For $p = \frac{13 + 7}{4} = \frac{20}{4} = 5$ - For $p = \frac{13 - 7}{4} = \frac{6}{4} = 1.5$ 10. **Find corresponding $q$ values using $q = 10 - 4p$:** - If $p=5$, then $q = 10 - 4 \times 5 = 10 - 20 = -10$ - If $p=1.5$, then $q = 10 - 4 \times 1.5 = 10 - 6 = 4$ 11. **Check that $p, q \neq 0$ for both solutions:** - $(p, q) = (5, -10)$ valid - $(p, q) = (1.5, 4)$ valid **Final answer:** $$\boxed{(p, q) = (5, -10) \text{ or } (1.5, 4)}$$