Cassandra started biking to the study group traveling 9 mph, after some time the bike got a flat so Cassandra walked the rest of the way, traveling 6 mph. If the total trip to the study group took 6 hours and it was 42 miles away, how long did Cassandra travel at each speed?

We can solve this problem by setting up a system of equations based on the information provided.

Let:
- \( t_1 \) be the time Cassandra spent biking (at 9 mph),
- \( t_2 \) be the time Cassandra spent walking (at 6 mph).

### Step 1: Create two equations from the given information

**Distance Equation:**
The total distance Cassandra traveled was 42 miles. So, the distance Cassandra biked plus the distance she walked must equal 42 miles.

\[
9t_1 + 6t_2 = 42
\]

**Time Equation:**
The total time spent traveling was 6 hours. So, the time spent biking plus the time spent walking must equal 6 hours.

\[
t_1 + t_2 = 6
\]

### Step 2: Solve the system of equations

First, solve the time equation for one variable, say \( t_2 \):

\[
t_2 = 6 - t_1
\]

Substitute this into the distance equation:

\[
9t_1 + 6(6 - t_1) = 42
\]

### Step 3: Simplify and solve for \( t_1 \)

Expand the equation:

\[
9t_1 + 36 - 6t_1 = 42
\]

Simplify:

\[
3t_1 + 36 = 42
\]

Subtract 36 from both sides:

\[
3t_1 = 6
\]

Solve for \( t_1 \):

\[
t_1 = 2
\]

### Step 4: Solve for \( t_2 \)

Now that we know \( t_1 = 2 \), substitute it into the equation for \( t_2 \):

\[
t_2 = 6 - 2 = 4
\]

### Conclusion:
- Cassandra biked for 2 hours.
- Cassandra walked for 4 hours.

Would you like more details on how to approach this type of problem, or do you have any questions?

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Here are 5 related questions you might find useful:
1. What if Cassandra biked at 10 mph instead of 9 mph? How would the times change?
2. How far did Cassandra bike and how far did she walk during her trip?
3. What would the total time be if Cassandra could walk faster at 8 mph?
4. If Cassandra took a break during her walk, how would that affect the total travel time?
5. What if Cassandra biked at a variable speed instead of a constant 9 mph?

**Tip:** When solving problems involving distance, speed, and time, always remember the formula \( \text{Distance} = \text{Speed} \times \text{Time} \). It helps set up the core equations!

Find out how long Cassandra biked and walked during her trip using algebraic methods. This problem involves systems of equations and the distance-speed-time relationship. Perfect for students in Grades 8-10.

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