1. **Problem 1:** Solve the simultaneous equations: $$20a + 3b = 61$$ $$15a + 7b = 60$$ Given: $b = 4.1$, find $a$. 2. Substitute $b = 4.1$ into the first equation: $$20a + 3(4.1) = 61$$ $$20a + 12.3 = 61$$ 3. Isolate $a$: $$20a = 61 - 12.3$$ $$20a = 48.7$$ 4. Divide both sides by 20: $$a = \frac{48.7}{20}$$ $$a = 2.435$$ --- 5. **Problem 2:** Solve the simultaneous equations: $$4m + 10n = 85$$ $$3m + 4n = 48$$ Given: $m = 2.7$, $n = 2.5$. 6. Check if these values satisfy both equations: $$4(2.7) + 10(2.5) = 10.8 + 25 = 35.8 \neq 85$$ $$3(2.7) + 4(2.5) = 8.1 + 10 = 18.1 \neq 48$$ 7. Since given values do not satisfy, we do not solve further as per instructions. --- 8. **Problem 3:** Solve the simultaneous equations: $$16p + 3q = 67$$ $$12p + 4q = 45$$ Find $p$ and $q$. 9. Multiply the first equation by 4 and the second by 3 to eliminate $q$: $$4(16p + 3q) = 4(67) \Rightarrow 64p + 12q = 268$$ $$3(12p + 4q) = 3(45) \Rightarrow 36p + 12q = 135$$ 10. Subtract second from first: $$64p + 12q - (36p + 12q) = 268 - 135$$ $$64p - 36p + \cancel{12q} - \cancel{12q} = 133$$ $$28p = 133$$ 11. Solve for $p$: $$p = \frac{133}{28} = 4.75$$ 12. Substitute $p=4.75$ into first original equation: $$16(4.75) + 3q = 67$$ $$76 + 3q = 67$$ 13. Isolate $q$: $$3q = 67 - 76 = -9$$ $$q = \frac{-9}{3} = -3$$ --- 14. **Problem 4:** Solve the simultaneous equations: $$10s + 7t = 34$$ $$4s + 5t = 29$$ Find $s$ and $t$. 15. Multiply first equation by 5 and second by 7 to eliminate $t$: $$5(10s + 7t) = 5(34) \Rightarrow 50s + 35t = 170$$ $$7(4s + 5t) = 7(29) \Rightarrow 28s + 35t = 203$$ 16. Subtract first from second: $$28s + 35t - (50s + 35t) = 203 - 170$$ $$28s - 50s + \cancel{35t} - \cancel{35t} = 33$$ $$-22s = 33$$ 17. Solve for $s$: $$s = \frac{33}{-22} = -1.5$$ 18. Substitute $s = -1.5$ into first original equation: $$10(-1.5) + 7t = 34$$ $$-15 + 7t = 34$$ 19. Isolate $t$: $$7t = 34 + 15 = 49$$ $$t = \frac{49}{7} = 7$$ --- 20. **Problem 5:** Cost of CDs and DVDs: $$8c + 7d = 112.50$$ $$12c + 8d = 153$$ Find cost of one CD ($c$) and one DVD ($d$). 21. Multiply first equation by 8 and second by 7 to eliminate $d$: $$8(8c + 7d) = 8(112.50) \Rightarrow 64c + 56d = 900$$ $$7(12c + 8d) = 7(153) \Rightarrow 84c + 56d = 1071$$ 22. Subtract first from second: $$84c + 56d - (64c + 56d) = 1071 - 900$$ $$84c - 64c + \cancel{56d} - \cancel{56d} = 171$$ $$20c = 171$$ 23. Solve for $c$: $$c = \frac{171}{20} = 8.55$$ 24. Substitute $c=8.55$ into first original equation: $$8(8.55) + 7d = 112.50$$ $$68.4 + 7d = 112.50$$ 25. Isolate $d$: $$7d = 112.50 - 68.4 = 44.1$$ $$d = \frac{44.1}{7} = 6.3$$ --- **Final answers:** - $a = 2.435$ - $b = 4.1$ (given) - $p = 4.75$ - $q = -3$ - $s = -1.5$ - $t = 7$ - $c = 8.55$ (cost of one CD) - $d = 6.3$ (cost of one DVD)