Candle A and Candle B are lit at the same time and both burn at a constant rate of 0.9 inches per hour.
For each candle we track the candle's remaining length in inches, $l$, in terms of the number of hours $t$ since the candle was lit.
a. If Candle $\mathrm{A}$ is initially 8 inches long and Candle B is initially 8 inches long, how many solutions will there be for this system? Enter "oo" (two o's) if there are infinitely many solutions.
solutions
b. If Candle A is initially 5 inches long and Candle B is initially 7 inches long, how many solutions will there be for this system? Enter "oo" (two o's) if there are infinitely many solutions.
solutions
Answer
Final Answer: For part a, there are infinitely many solutions, represented as \(\boxed{oo}\). For part b, there are no solutions, represented as \(\boxed{0}\).
Step 1 ：The question is asking for the number of solutions to the system of equations representing the lengths of the candles over time. Since both candles burn at the same rate, the equations for their lengths over time are linear and have the same slope. The number of solutions to the system depends on whether the y-intercepts of the two lines (the initial lengths of the candles) are the same or different.
Step 2 ：For part a, both candles start at 8 inches, so the equations for their lengths over time are the same, meaning there are infinitely many solutions.
Step 3 ：For part b, the candles start at different lengths, so the equations for their lengths over time are different lines with the same slope. Since they are parallel and do not intersect, there are no solutions to the system.
Step 4 ：Final Answer: For part a, there are infinitely many solutions, represented as \(\boxed{oo}\). For part b, there are no solutions, represented as \(\boxed{0}\).
