Linear Systems Word 764717

1. **Problem a:** George invested $2000 in two accounts. One earned 4.5% interest, the other lost 2%, with a net gain of $38. Find the amount invested in each account. 2. **Formulas and variables:** Let $x$ be the amount invested at 4.5% and $y$ the amount invested at -2%. Total investment: $$x + y = 2000$$ Interest gained: $$0.045x - 0.02y = 38$$ 3. **Solve the system:** From the first equation, $$y = 2000 - x$$ Substitute into the second: $$0.045x - 0.02(2000 - x) = 38$$ 4. **Simplify:** $$0.045x - 40 + 0.02x = 38$$ $$0.065x - 40 = 38$$ $$0.065x = 78$$ 5. **Calculate $x$:** $$x = \frac{78}{0.065}$$ $$x = 1200$$ 6. **Calculate $y$:** $$y = 2000 - 1200 = 800$$ 7. **Answer a:** $1200$ invested at 4.5%, $800$ invested at -2%. --- 1. **Problem b:** Martha mixes 10% fat cream and 2% fat cream to make 30 mL of 5% fat cream. Find amounts mixed. 2. **Variables:** Let $x$ be mL of 10% cream, $y$ be mL of 2% cream. Total volume: $$x + y = 30$$ Fat content: $$0.10x + 0.02y = 0.05 \times 30 = 1.5$$ 3. **Solve for $y$:** $$y = 30 - x$$ Substitute: $$0.10x + 0.02(30 - x) = 1.5$$ 4. **Simplify:** $$0.10x + 0.6 - 0.02x = 1.5$$ $$0.08x + 0.6 = 1.5$$ $$0.08x = 0.9$$ 5. **Calculate $x$:** $$x = \frac{0.9}{0.08} = 11.25$$ 6. **Calculate $y$:** $$y = 30 - 11.25 = 18.75$$ 7. **Answer b:** $11.25$ mL of 10% cream and $18.75$ mL of 2% cream. --- 1. **Problem c:** Canoe travels 10 km upstream in 40 min and downstream in 25 min. Find canoe speed in still water ($c$) and current speed ($w$). 2. **Variables:** Let $c$ = canoe speed (km/h), $w$ = current speed (km/h). Convert times to hours: 40 min = $\frac{2}{3}$ h, 25 min = $\frac{5}{12}$ h. 3. **Equations:** Upstream speed: $c - w$, time: $\frac{2}{3}$ h Downstream speed: $c + w$, time: $\frac{5}{12}$ h Distance = speed $\times$ time = 10 km 4. **Write equations:** $$ (c - w) \times \frac{2}{3} = 10 $$ $$ (c + w) \times \frac{5}{12} = 10 $$ 5. **Simplify:** $$ c - w = \frac{10}{(2/3)} = 10 \times \frac{3}{2} = 15 $$ $$ c + w = \frac{10}{(5/12)} = 10 \times \frac{12}{5} = 24 $$ 6. **Solve system:** Add equations: $$ (c - w) + (c + w) = 15 + 24 $$ $$ 2c = 39 $$ $$ c = 19.5 $$ 7. **Find $w$:** $$ 19.5 + w = 24 $$ $$ w = 24 - 19.5 = 4.5 $$ 8. **Answer c:** Canoe speed is $19.5$ km/h, current speed is $4.5$ km/h. --- 1. **Problem d:** Bag has 1234 coins, quarters and dimes, total value 290.05. Find number of each coin. 2. **Variables:** Let $q$ = quarters, $d$ = dimes. Total coins: $$ q + d = 1234 $$ Total value in cents: $$ 25q + 10d = 29005 $$ 3. **Solve for $d$:** $$ d = 1234 - q $$ Substitute: $$ 25q + 10(1234 - q) = 29005 $$ 4. **Simplify:** $$ 25q + 12340 - 10q = 29005 $$ $$ 15q + 12340 = 29005 $$ $$ 15q = 16665 $$ 5. **Calculate $q$:** $$ q = \frac{16665}{15} = 1111 $$ 6. **Calculate $d$:** $$ d = 1234 - 1111 = 123 $$ 7. **Answer d:** $1111$ quarters and $123$ dimes.