1. Solve the proportion $\frac{3a+8}{6} = \frac{a-7}{9}$. Use cross multiplication: $9(3a+8) = 6(a-7)$. Expand both sides: $27a + 72 = 6a - 42$. Bring all terms involving $a$ to one side: $27a - 6a = -42 - 72$. Simplify: $21a = -114$. Divide both sides by 21: $$a = \frac{-114}{21}$$. Show cancellation: $$a = \frac{\cancel{-114}}{\cancel{21}} = -\frac{114}{21} = -\frac{38}{7}$$. Final answer: $a = -\frac{38}{7}$. 2. Solve the proportion $\frac{3n-5}{2n+7} = \frac{9}{2}$. Cross multiply: $2(3n-5) = 9(2n+7)$. Expand: $6n - 10 = 18n + 63$. Bring terms with $n$ to one side: $6n - 18n = 63 + 10$. Simplify: $-12n = 73$. Divide both sides by $-12$: $$n = \frac{73}{-12} = -\frac{73}{12}$$. 3. Solve the proportion $\frac{a}{4} = \frac{-8}{6}$. Cross multiply: $6a = 4(-8)$. Simplify right side: $6a = -32$. Divide both sides by 6: $$a = \frac{-32}{6}$$. Show cancellation: $$a = \frac{\cancel{-32}}{\cancel{6}} = -\frac{16}{3}$$. 4. Solve the proportion $\frac{10}{8} = \frac{4}{a}$. Cross multiply: $10a = 8 \times 4$. Simplify right side: $10a = 32$. Divide both sides by 10: $$a = \frac{32}{10}$$. Show cancellation: $$a = \frac{\cancel{32}}{\cancel{10}} = \frac{16}{5}$$. 5. Find $\sin Z$ in triangle $ZXY$ with opposite side 8 and hypotenuse 17. Formula: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$. Calculate: $\sin Z = \frac{8}{17}$. 6. Find $\tan A$ in triangle $ABC$ with opposite side 25 and adjacent side 7. Formula: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. Calculate: $\tan A = \frac{25}{7}$. 7. Find $\cos C$ in triangle $ABC$ with adjacent side 21 and hypotenuse 35. Formula: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. Calculate: $\cos C = \frac{21}{35}$. Show cancellation: $\cos C = \frac{\cancel{21}}{\cancel{35}} = \frac{3}{5}$. 8. Find $\cos C$ in triangle $ABC$ with adjacent side 28 and hypotenuse 35. Calculate: $\cos C = \frac{28}{35}$. Show cancellation: $\cos C = \frac{\cancel{28}}{\cancel{35}} = \frac{4}{5}$. 9. Find $\sin X$ in triangle $ZXY$ with opposite side 15 and hypotenuse 17. Calculate: $\sin X = \frac{15}{17}$. 10. Find $\tan X$ in triangle $ZXY$ with opposite side 8 and adjacent side 15. Calculate: $\tan X = \frac{8}{15}$. 11. Find missing side $x$ in right triangle with angle 60°, adjacent side 16, hypotenuse $x$. Use $\cos 60^\circ = \frac{\text{adjacent}}{\text{hypotenuse}}$. Calculate: $\cos 60^\circ = \frac{16}{x}$. Since $\cos 60^\circ = \frac{1}{2}$, set $\frac{1}{2} = \frac{16}{x}$. Cross multiply: $x = 32$. 12. Find missing side $x$ in right triangle with sides 15 (adjacent), 16 (hypotenuse), and $x$ (opposite). Use Pythagorean theorem: $x^2 + 15^2 = 16^2$. Calculate: $x^2 + 225 = 256$. Subtract 225: $x^2 = 31$. Take square root: $x = \sqrt{31} \approx 5.6$. 13. Find missing side $x$ in right triangle with side 12 (adjacent), angle 74°, hypotenuse $x$. Use $\cos 74^\circ = \frac{12}{x}$. Calculate $\cos 74^\circ \approx 0.2756$. Set $0.2756 = \frac{12}{x}$. Cross multiply: $x = \frac{12}{0.2756} \approx 43.5$. 14. Find missing side $x$ in right triangle with side 17 (adjacent), angle 60°, hypotenuse $x$. Use $\cos 60^\circ = \frac{17}{x}$. Since $\cos 60^\circ = \frac{1}{2}$, set $\frac{1}{2} = \frac{17}{x}$. Cross multiply: $x = 34$. 15. Find angle $?$ in right triangle with sides 33 (adjacent), 41 (hypotenuse). Use $\cos ? = \frac{33}{41}$. Calculate $\cos ? \approx 0.8049$. Find angle: $? = \cos^{-1}(0.8049) \approx 36^\circ$. 16. Find angle $?$ in right triangle with sides 40 (hypotenuse), 23 (opposite). Use $\sin ? = \frac{23}{40}$. Calculate $\sin ? = 0.575$. Find angle: $? = \sin^{-1}(0.575) \approx 35^\circ$. 17. Find angle $?$ in right triangle with sides 18 (adjacent), 7 (opposite). Use $\tan ? = \frac{7}{18}$. Calculate $\tan ? \approx 0.3889$. Find angle: $? = \tan^{-1}(0.3889) \approx 21^\circ$. 18. Find angle $?$ in right triangle with sides 59 (hypotenuse), 25 (opposite). Use $\sin ? = \frac{25}{59}$. Calculate $\sin ? \approx 0.4237$. Find angle: $? = \sin^{-1}(0.4237) \approx 25^\circ$. 19. Identify center and radius of circle $(x-4)^2 + (y-1)^2 = 3$. Center: $(4,1)$. Radius: $\sqrt{3} \approx 1.732$. 20. Identify center and radius of circle $x^2 + y^2 + 8x - 6y + 16 = 0$. Complete the square: Group $x$: $x^2 + 8x$, group $y$: $y^2 - 6y$, constant $+16$. Complete square for $x$: $x^2 + 8x + 16$ (add 16). Complete square for $y$: $y^2 - 6y + 9$ (add 9). Add 16 + 9 = 25 to left side, so add 25 to right side to keep equality. Rewrite: $(x+4)^2 + (y-3)^2 = 25 - 16 = 9$. Center: $(-4,3)$. Radius: $\sqrt{9} = 3$. 21. Factor $3n^2 + 25n - 18$. Find factors of $3 \times (-18) = -54$ that sum to 25: 27 and -2. Rewrite: $3n^2 + 27n - 2n - 18$. Group: $3n(n+9) - 2(n+9)$. Factor: $(3n - 2)(n + 9)$. 22. Factor $7x^2 - 8x - 12$. Find factors of $7 \times (-12) = -84$ that sum to -8: 6 and -14. Rewrite: $7x^2 + 6x - 14x - 12$. Group: $x(7x + 6) - 2(7x + 6)$. Factor: $(x - 2)(7x + 6)$. 23. Sketch graph of $f(x) = -(x-2)^2 + 2$. This is a parabola opening downward with vertex at $(2,2)$. 24. Sketch graph of $f(x) = -(x+2)^2 + 3$. This is a parabola opening downward with vertex at $(-2,3)$.