question_answer 30)

                Simplify and solve the following linear equations: 7.            \[3(t-3)=5(2t+1)\] 8.            \[15(y-4)-2(y-9)+5(y+6)=0\] 9.            \[3(5z-7)\,-2(9z-11)=4(8z-13)-17\] 10.          \[0.25\,(4f-3)=0.05(10f-9)\]

Answer:

                7. \[3\,(t-3)=5(2t+1)\]                 We have \[3\,(t-3)=5(2t+1)\]      |Transposing 10t to RHS and ? 9 to RHS \[\Rightarrow \]               \[-7t=14\] \[\Rightarrow \]               \[t=-\frac{14}{7}=-2\]                                     |Dividing both sides by ? 7 This is the required solution. Check: LHS \[=3(t-3)\] \[=3(-2-3)\]\[=3(-5)\] \[=-15\]                RHS \[=5(2t+1)\] \[=5(2\times (-2)+1)\] \[=5(-4+1)\] \[=5(-3)\] \[=-15\] 8.            \[15(y-4)-2(y-9)+5(y+6)=0\] We have \[15(y-4)-2(y-9)\]\[+5(y+6)=0\] \[\Rightarrow \]               \[15y-60-2y+18+5y+30=0\] \[\Rightarrow \]               \[18y-12=0\] \[\Rightarrow \]               \[18y=12\]                                          |Transposing ? 12 to RHS \[\Rightarrow \]               \[y=\frac{12}{18}=\frac{2}{3}\]                                  |Dividing both sides by 18 This is the required solution. Check: LHS \[=15(y-4)-2(y-9)\]\[+5(y+6)\] \[=15\left( \frac{2}{3}-4 \right)-2\left( \frac{2}{3}-9 \right)\,+5\left( \frac{2}{3}+6 \right)\] \[=15\left( \frac{2-12}{3} \right)-2\,\left( \frac{2-27}{3} \right)+5\left( \frac{2+18}{3} \right)\] \[=15\left( -\frac{10}{3} \right)\,-2\left( -\frac{25}{3} \right)\,+5\left( \frac{20}{3} \right)\] \[=-50+\frac{50}{3}+\frac{100}{3}\] \[=\frac{-150+50+100}{3}=\frac{0}{3}=0\] = RHS (as required). 9.            \[3(5z-7)-2(9z-11)=4(8z-13)-17\] We have \[3(5z-7)-2(9z-11)=4(8z-13)-17\] \[\Rightarrow \]               \[15z-21-18z+22=32z-52-17\] \[\Rightarrow \]               \[-3z+1=32z-69\] \[\Rightarrow \]               \[-3z-32z=-69-1\]             |Transposing 32z to LHS and 1 to RHS \[\Rightarrow \]               \[-35z=-70\] \[\Rightarrow \]               \[z=\frac{-70}{-35}=2\]                  |Dividing both sides by ? 35 This is the required solution. Check: LHS \[=3(5z-7)-2(9z-11)\] \[=3(5\times 2-7)\]\[-2(9\times 2-11)\] \[=3(10-7)-2(18-11)\] \[=3(3)-2(7)\] \[=9-14\] \[=-5\] RHS \[=4(8z-13)-17\] \[=4(8\times 2-13)-17\] \[=4(16-13)-17\] \[=4(3)-17\] \[=12-17\] \[=-5\] = LHS (as required). 10.          \[0.25(4f-3)=0.05\,(10f-9)\]                 We have \[0.25(4f-3)=0.05\,(10f-9)\] \[\Rightarrow \]               \[f-0.75=0.5f-0.45\] \[\Rightarrow \]               \[f-0.5f=-0.45+0.75\]      |Transposing 0.5f to LHS and ? 0.75 to RHS \[\Rightarrow \]               \[0.5f=0.30\] \[\Rightarrow \]               \[f=\frac{0.30}{0.5}=0.6\]                             | Dividing both sides by 0.5 This is the required solution. Check: LHS          \[=0.25\,(4f-3)\] \[=0.25(4\times 0.6-3)\] \[=0.25(2.4-3)\] \[=0.25(-0.6)\] \[=-0.15\]                                 RHS        \[=0.05\,(10f-9)\] \[=0.05(10\times 0.6-9)\] \[=0.05(6-9)\] \[=0.05(-3)\] \[=-0.15\]                                                 = LHS (as required).