1. **State the problem:** Solve the system of equations: $$\sqrt{x} + 5z = 18$$ $$\frac{y^2 - 2}{x} = 7$$ $$z^2 - y = 31$$ 2. **Rewrite the equations for clarity:** Equation 1: $$\sqrt{x} + 5z = 18$$ Equation 2: $$\frac{y^2 - 2}{x} = 7$$ Equation 3: $$z^2 - y = 31$$ 3. **Express variables where possible:** From Equation 3: $$y = z^2 - 31$$ 4. **Substitute $y$ into Equation 2:** $$\frac{(z^2 - 31)^2 - 2}{x} = 7$$ Multiply both sides by $x$: $$ (z^2 - 31)^2 - 2 = 7x $$ 5. **Express $x$ in terms of $z$:** $$ x = \frac{(z^2 - 31)^2 - 2}{7} $$ 6. **Substitute $x$ into Equation 1:** $$ \sqrt{\frac{(z^2 - 31)^2 - 2}{7}} + 5z = 18 $$ 7. **Isolate the square root:** $$ \sqrt{\frac{(z^2 - 31)^2 - 2}{7}} = 18 - 5z $$ 8. **Square both sides:** $$ \frac{(z^2 - 31)^2 - 2}{7} = (18 - 5z)^2 $$ Multiply both sides by 7: $$ (z^2 - 31)^2 - 2 = 7(18 - 5z)^2 $$ 9. **Expand both sides:** Left side: $$ (z^2 - 31)^2 - 2 = (z^2)^2 - 2 \cdot 31 z^2 + 31^2 - 2 = z^4 - 62 z^2 + 961 - 2 = z^4 - 62 z^2 + 959 $$ Right side: $$ 7(18 - 5z)^2 = 7(324 - 180z + 25z^2) = 2268 - 1260 z + 175 z^2 $$ 10. **Set equation:** $$ z^4 - 62 z^2 + 959 = 2268 - 1260 z + 175 z^2 $$ Bring all terms to one side: $$ z^4 - 62 z^2 + 959 - 2268 + 1260 z - 175 z^2 = 0 $$ Simplify: $$ z^4 - (62 + 175) z^2 + 1260 z + (959 - 2268) = 0 $$ $$ z^4 - 237 z^2 + 1260 z - 1309 = 0 $$ 11. **Solve quartic equation:** This quartic is complex; try rational roots by testing factors of 1309. Test $z=7$: $$7^4 - 237 \cdot 7^2 + 1260 \cdot 7 - 1309 = 2401 - 237 \cdot 49 + 8820 - 1309$$ Calculate: $$237 \cdot 49 = 11613$$ So: $$2401 - 11613 + 8820 - 1309 = (2401 - 11613) + (8820 - 1309) = -9212 + 7511 = -1701 \neq 0$$ Try $z=13$: $$13^4 - 237 \cdot 13^2 + 1260 \cdot 13 - 1309 = 28561 - 237 \cdot 169 + 16380 - 1309$$ Calculate: $$237 \cdot 169 = 40053$$ So: $$28561 - 40053 + 16380 - 1309 = (28561 - 40053) + (16380 - 1309) = -11492 + 15071 = 3579 \neq 0$$ Try $z=1$: $$1 - 237 + 1260 - 1309 = 1 - 237 + 1260 - 1309 = -285 \neq 0$$ Try $z=3$: $$81 - 237 \cdot 9 + 3780 - 1309 = 81 - 2133 + 3780 - 1309 = 419 \neq 0$$ Try $z=9$: $$6561 - 237 \cdot 81 + 11340 - 1309 = 6561 - 19197 + 11340 - 1309 = -2605 \neq 0$$ Try $z=11$: $$14641 - 237 \cdot 121 + 13860 - 1309 = 14641 - 28677 + 13860 - 1309 = -1485 \neq 0$$ Try $z=14$: $$38416 - 237 \cdot 196 + 17640 - 1309 = 38416 - 46452 + 17640 - 1309 = 2295 \neq 0$$ Try $z=10$: $$10000 - 237 \cdot 100 + 12600 - 1309 = 10000 - 23700 + 12600 - 1309 = -4409 \neq 0$$ Try $z=15$: $$50625 - 237 \cdot 225 + 18900 - 1309 = 50625 - 53325 + 18900 - 1309 = 1891 \neq 0$$ Try $z=5$: $$625 - 237 \cdot 25 + 6300 - 1309 = 625 - 5925 + 6300 - 1309 = -309 \neq 0$$ Try $z=6$: $$1296 - 237 \cdot 36 + 7560 - 1309 = 1296 - 8532 + 7560 - 1309 = 1015 \neq 0$$ Try $z=8$: $$4096 - 237 \cdot 64 + 10080 - 1309 = 4096 - 15168 + 10080 - 1309 = -301 \neq 0$$ No simple integer roots found; approximate numerically. 12. **Numerical approximation:** Try $z=4$: $$256 - 237 \cdot 16 + 5040 - 1309 = 256 - 3792 + 5040 - 1309 = 195$$ Try $z=3.5$: $$150.06 - 237 \cdot 12.25 + 4410 - 1309 = 150.06 - 2900.25 + 4410 - 1309 = 350.81$$ Try $z=2$: $$16 - 237 \cdot 4 + 2520 - 1309 = 16 - 948 + 2520 - 1309 = 279$$ Try $z=0$: $$0 - 0 + 0 - 1309 = -1309$$ Try $z=1.5$: $$5.06 - 237 \cdot 2.25 + 1890 - 1309 = 5.06 - 533.25 + 1890 - 1309 = 52.81$$ Try $z=1$: $$1 - 237 + 1260 - 1309 = -285$$ Try $z=1.2$: $$2.07 - 237 \cdot 1.44 + 1512 - 1309 = 2.07 - 341.28 + 1512 - 1309 = -136.21$$ Try $z=1.3$: $$2.85 - 237 \cdot 1.69 + 1638 - 1309 = 2.85 - 400.53 + 1638 - 1309 = -68.68$$ Try $z=1.4$: $$3.84 - 237 \cdot 1.96 + 1764 - 1309 = 3.84 - 464.52 + 1764 - 1309 = -5.68$$ Try $z=1.45$: $$4.42 - 237 \cdot 2.10 + 1827 - 1309 = 4.42 - 497.7 + 1827 - 1309 = 24.72$$ Try $z=1.42$: $$4.07 - 237 \cdot 2.02 + 1789 - 1309 = 4.07 - 478.74 + 1789 - 1309 = 9.33$$ Try $z=1.41$: $$3.95 - 237 \cdot 1.99 + 1775 - 1309 = 3.95 - 471.63 + 1775 - 1309 = 1.82$$ Try $z=1.405$: $$3.89 - 237 \cdot 1.98 + 1769 - 1309 = 3.89 - 468.08 + 1769 - 1309 = -1.93$$ Try $z=1.4075$: $$3.92 - 237 \cdot 1.99 + 1772 - 1309 = 3.92 - 469.85 + 1772 - 1309 = -0.05$$ Try $z=1.408$: $$3.93 - 237 \cdot 1.99 + 1773 - 1309 = 3.93 - 470.12 + 1773 - 1309 = 0.88$$ So root near $z \approx 1.4075$. 13. **Calculate $y$ and $x$:** $$ y = z^2 - 31 = (1.4075)^2 - 31 = 1.981 - 31 = -29.019 $$ $$ x = \frac{(z^2 - 31)^2 - 2}{7} = \frac{(-29.019)^2 - 2}{7} = \frac{841.11 - 2}{7} = \frac{839.11}{7} = 119.87 $$ 14. **Check Equation 1:** $$ \sqrt{x} + 5z = \sqrt{119.87} + 5 \times 1.4075 = 10.95 + 7.04 = 17.99 \approx 18 $$ 15. **Final solution:** $$ x \approx 119.87, \quad y \approx -29.02, \quad z \approx 1.41 $$