Algebra 1 Topic 2: Linear Equations NJGPA REVIEW Part B What equation expresses \( P \), the amount in dollars, of Melissa's mother's donation as a function of \( M \), the number of miles that Melissa walks? A) \( P=M+10 \) B) \( P=2 M+10 \) C) \( P=10 M+2 \) D) \( P=2(M+10) \) Part C If the donation is \( \$ 44 \), how many miles did Melissa walk? A) 15 B) 17 C) 22 D) 98 Part D Melissa wants her mother to donate at least \( \$ 65 \). Which of the listed numbers of miles would make the donation at least \( \$ 65 \) ? Select all that apply. A) 23 miles B) 25 miles C) 27 miles D) 29 miles E) 31 miles F) 33 miles Example 6: Use the given information to answer Part A and Part B. Part A A dump truck weighs 11.25 tons when empty. A conveyor belt pours sand into the truck at a constant rate of \( \frac{1}{4} \) ton per minute until it is full. Let \( t \) represent the elapsed time in minutes. Let \( w \) represent the weight of the truck after \( t \) minutes. Write a linear equation for \( w \) in terms of \( t \). Part B The dump truck from Part A weighs 18 tons when filled. At the same time the dump truck is being filled an identical dump truck filled to capacity is being emptied at a rate of \( \frac{1}{8} \) ton per minute. How much sand is in each dump truck when the trucks are the same weight? 22

Sure, let's work through each part of the Algebra 1 Linear Equations review step by step. --- ### **Melissa's Donation Problem** #### **Part B** **Question:** What equation expresses \( P \), the amount in dollars, of Melissa's mother's donation as a function of \( M \), the number of miles that Melissa walks? **Options:** A) \( P = M + 10 \) B) \( P = 2M + 10 \) C) \( P = 10M + 2 \) D) \( P = 2(M + 10) \) **Solution:** Typically, such problems involve a base donation plus an amount per mile walked. Here, it appears that the donation increases by \$2 for each mile walked, starting with a base amount of \$10. Thus, the correct equation is: \[ P = 2M + 10 \] **Answer:** **B) \( P = 2M + 10 \)** --- #### **Part C** **Question:** If the donation is \$44, how many miles did Melissa walk? **Options:** A) 15 B) 17 C) 22 D) 98 **Solution:** Using the equation from Part B: \[ 44 = 2M + 10 \] Subtract 10 from both sides: \[ 34 = 2M \] Divide both sides by 2: \[ M = 17 \] **Answer:** **B) 17** --- #### **Part D** **Question:** Melissa wants her mother to donate at least \$65. Which of the listed numbers of miles would make the donation at least \$65? **Select all that apply.** **Options:** A) 23 miles B) 25 miles C) 27 miles D) 29 miles E) 31 miles F) 33 miles **Solution:** Using the equation \( P = 2M + 10 \), set \( P \geq 65 \): \[ 2M + 10 \geq 65 \] Subtract 10 from both sides: \[ 2M \geq 55 \] Divide by 2: \[ M \geq 27.5 \] Since the number of miles must be a whole number, \( M \) must be **28** or greater. **Applicable Options:** - **D) 29 miles** - **E) 31 miles** - **F) 33 miles** **Answers:** **D), E), and F)** --- ### **Example 6: Dump Truck Problem** #### **Part A** **Question:** A dump truck weighs 11.25 tons when empty. A conveyor belt pours sand into the truck at a constant rate of \( \frac{1}{4} \) ton per minute until it is full. Let \( t \) represent the elapsed time in minutes. Let \( w \) represent the weight of the truck after \( t \) minutes. Write a linear equation for \( w \) in terms of \( t \). **Solution:** The weight of the truck increases by \( \frac{1}{4} \) ton each minute. Starting from 11.25 tons, the equation is: \[ w = \frac{1}{4}t + 11.25 \] **Answer:** \[ w = \frac{1}{4}t + 11.25 \] #### **Part B** **Question:** The dump truck from Part A weighs 18 tons when filled. At the same time the dump truck is being filled, an identical dump truck filled to capacity is being emptied at a rate of \( \frac{1}{8} \) ton per minute. How much sand is in each dump truck when the trucks are the same weight? **Solution:** Let's define: - **Truck 1:** Being filled from empty at \( \frac{1}{4} \) ton/min. - **Truck 2:** Being emptied from full (18 tons) at \( \frac{1}{8} \) ton/min. Let \( t \) be the time in minutes until both trucks weigh the same. For Truck 1: \[ w_1 = \frac{1}{4}t + 11.25 \] For Truck 2: \[ w_2 = 18 - \frac{1}{8}t \] Set \( w_1 = w_2 \): \[ \frac{1}{4}t + 11.25 = 18 - \frac{1}{8}t \] Multiply every term by 8 to eliminate fractions: \[ 2t + 90 = 144 - t \] Combine like terms: \[ 3t = 54 \] \[ t = 18 \] minutes Now, find the weight at \( t = 18 \) minutes: \[ w_1 = \frac{1}{4}(18) + 11.25 = 4.5 + 11.25 = 15.75 \] tons \[ w_2 = 18 - \frac{1}{8}(18) = 18 - 2.25 = 15.75 \] tons **Amount of Sand in Each Truck:** Since the empty truck weighs 11.25 tons: \[ \text{Sand in each truck} = 15.75 - 11.25 = 4.5 \] tons **Answer:** Each dump truck contains **4.5 tons** of sand when they are the same weight. --- ### **Summary of Answers** - **Melissa's Donation Problem:** - **Part B:** **B) \( P = 2M + 10 \)** - **Part C:** **B) 17** - **Part D:** **D) 29 miles**, **E) 31 miles**, **F) 33 miles** - **Example 6: Dump Truck Problem:** - **Part A:** \( w = \frac{1}{4}t + 11.25 \) - **Part B:** Each truck has **4.5 tons** of sand when they weigh the same. Feel free to ask if you need further clarification on any of these problems!