Complete the following word problems using systems of equations. 1. Astronauts traveling to the International Space Station have to be very careful about weight (it takes about 9 lbs of rocket fuel to send 1 lb of stuff to the station). Each astronaut can bring approximately 11 lbs of personal items (mementos and clothes) with them into space. Each memento weighs about 0.1 lb , and each item of clothing weighs about 0.5 Ibs . If an astronaut wanted to bring 30 items with them, how many mementos and pieces of clothing would they bring?

Let's denote the number of mementos as \( m \) and the number of pieces of clothing as \( c \). We are given the following information: 1. Each memento weighs 0.1 lb. 2. Each piece of clothing weighs 0.5 lbs. 3. Each astronaut can bring approximately 11 lbs of personal items. 4. It takes about 9 lbs of rocket fuel to send 1 lb of stuff to the station. 5. The astronaut wants to bring 30 items with them. We can set up the following system of equations based on the given information: 1. The total weight of the mementos and clothing should be less than or equal to 11 lbs: \[ 0.1m + 0.5c \leq 11 \] 2. The total number of items should be 30: \[ m + c = 30 \] We can solve this system of equations to find the number of mementos and pieces of clothing the astronaut would bring. Solve the system of equations by following steps: - step0: Solution: \(\left\{ \begin{array}{l}0.1m+0.5c\leq 11\\m+c=30\end{array}\right.\) - step1: Solve the inequality: \(\left\{ \begin{array}{l}0.1m+0.5c-11\leq 0\\c+m=30\end{array}\right.\) - step2: Find the intersection: \(\left(c,m\right) = \left(20,10\right)\) - step3: Alternative Form: \(\textrm{Infinitely many solutions}\) The system of equations has infinitely many solutions, which means there are multiple combinations of mementos and pieces of clothing that the astronaut can bring to meet the given conditions. To find the specific combination of mementos and pieces of clothing, we can consider the constraints: 1. The total weight of the mementos and clothing should be less than or equal to 11 lbs. 2. The total number of items should be 30. We can try different combinations of mementos and pieces of clothing to find a valid solution that satisfies both conditions. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}0.1m+0.5c=11\\m+c=30\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}0.1m+0.5c=11\\c=30-m\end{array}\right.\) - step2: Substitute the value of \(c:\) \(0.1m+0.5\left(30-m\right)=11\) - step3: Simplify: \(-0.4m+15=11\) - step4: Move the constant to the right side: \(-0.4m=11-15\) - step5: Subtract the numbers: \(-0.4m=-4\) - step6: Change the signs: \(0.4m=4\) - step7: Divide both sides: \(\frac{0.4m}{0.4}=\frac{4}{0.4}\) - step8: Divide the numbers: \(m=10\) - step9: Substitute the value of \(m:\) \(c=30-10\) - step10: Simplify: \(c=20\) - step11: Calculate: \(\left\{ \begin{array}{l}c=20\\m=10\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}c=20\\m=10\end{array}\right.\) - step13: Rewrite: \(\left(c,m\right) = \left(20,10\right)\) The solution to the system of equations is \( c = 20 \) and \( m = 10 \). Therefore, the astronaut would bring 10 mementos and 20 pieces of clothing with them to meet the given conditions.