Questions: Kayden and Alondra both walk away from Kayden's house at a constant speed of 7 feet per second. For each person, we will track that person's distance from Kayden's house in feet, d, in terms of the number of seconds t since they started walking. a. If Kayden is initially 25 feet from his house and Alondra is initially 45 feet from Kayden's house, how many solutions will there be for this system? (Enter " 00 " if there are infinitely many solutions.) b. If Kayden is initially 15 feet from his house and Alondra is initially 15 feet from Kayden's house, how many solutions will there be for this system? (Enter " 00 " if there are infinitely many solutions.)

Kayden and Alondra both walk away from Kayden's house at a constant speed of 7 feet per second. For each person, we will track that person's distance from Kayden's house in feet, d, in terms of the number of seconds t since they started walking.
a. If Kayden is initially 25 feet from his house and Alondra is initially 45 feet from Kayden's house, how many solutions will there be for this system? (Enter " 00 " if there are infinitely many solutions.)

b. If Kayden is initially 15 feet from his house and Alondra is initially 15 feet from Kayden's house, how many solutions will there be for this system? (Enter " 00 " if there are infinitely many solutions.)

Transcript text: Kayden and Alondra both walk away from Kayden's house at a constant speed of 7 feet per second. For each person, we will track that person's distance from Kayden's house in feet, $d$, in terms of the number of seconds $t$ since they started walking. a. If Kayden is initially 25 feet from his house and Alondra is initially 45 feet from Kayden's house, how many solutions will there be for this system? (Enter " 00 " if there are infinitely many solutions.) $\square$ b. If Kayden is initially 15 feet from his house and Alondra is initially 15 feet from Kayden's house, how many solutions will there be for this system? (Enter " 00 " if there are infinitely many solutions.) $\square$

Solution

Solution Steps

To determine the number of solutions for each system, we need to set up equations for the distances of Kayden and Alondra from Kayden's house as functions of time and then find the points where these distances are equal.

a. For Kayden, the distance from his house is given by $ d_K = 25 + 7t $. For Alondra, the distance from Kayden's house is given by $ d_A = 45 + 7t $. We need to find the number of solutions to the equation $ 25 + 7t = 45 + 7t $.

b. For Kayden, the distance from his house is given by $ d_K = 15 + 7t $. For Alondra, the distance from Kayden's house is given by $ d_A = 15 + 7t $. We need to find the number of solutions to the equation $ 15 + 7t = 15 + 7t $.

Step 1: Analyze Part a

For Kayden, the distance from his house is given by: \[ d_K = 7t + 25 \] For Alondra, the distance from Kayden's house is: \[ d_A = 7t + 45 \] Setting these distances equal to find the solutions: \[ 7t + 25 = 7t + 45 \] This simplifies to: \[ 25 = 45 \] This equation is false, indicating that there are no solutions for this system.

Step 2: Analyze Part b

For Kayden, the distance from his house is given by: \[ d_K = 7t + 15 \] For Alondra, the distance from Kayden's house is: \[ d_A = 7t + 15 \] Setting these distances equal to find the solutions: \[ 7t + 15 = 7t + 15 \] This simplifies to: \[ 15 = 15 \] This equation is always true, indicating that there are infinitely many solutions for this system.

Final Answer

For part a, the number of solutions is $0$. For part b, the number of solutions is $00$ (indicating infinitely many solutions). Thus, the final answers are: \[ \boxed{0} \] \[ \boxed{00} \]

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