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The image presents a problem about two plans for the descent of lunar modules, with specific rates of descent and initial altitudes.

### Plan details:
- **Plan 1**: Starts at 100 thousand feet, descending at a rate of 0.5 thousand feet per minute.
- **Plan 2**: Starts at 150 thousand feet, descending at a rate of 1 thousand feet per minute.

The question asks to express this situation as a system of equations where \( y \) represents altitude (in thousands of feet) and \( x \) represents time (in minutes). The form of the equations is:

\[
\begin{aligned}
y &= a + bx \\
y &= c + dx
\end{aligned}
\]

We need to find the constants \(a\), \(b\), \(c\), and \(d\).

### Plan 1:
- Initial altitude: 100 thousand feet, so \(a = 100\).
- Descent rate: 0.5 thousand feet per minute, so \(b = -0.5\) (negative because altitude decreases over time).

### Plan 2:
- Initial altitude: 150 thousand feet, so \(c = 150\).
- Descent rate: 1 thousand feet per minute, so \(d = -1\).

Thus, the system of equations is:

\[
\begin{aligned}
y &= 100 - 0.5x \\
y &= 150 - x
\end{aligned}
\]

#### Final values:
- \( a = 100 \)
- \( b = -0.5 \)
- \( c = 150 \)
- \( d = -1 \)

Would you like more details or clarifications? 

Here are some related questions to expand your understanding:

1. How would the equations change if the descent rates were reversed?
2. What would the graph of both equations look like on the grid provided?
3. How long will it take for the module to reach the surface in both plans?
4. What does the intersection point of these two lines represent?
5. How can we find when the two plans will result in the same altitude?

**Tip**: When graphing linear equations, the slope (rate of change) gives you a quick sense of how fast the dependent variable (like altitude) changes with time.

When rockets approached the moon, engineers needed to plan how the lunar modules would land. They developed two plans. Plan 1: At 100 thousand feet, the module drops at a steady rate of 0.5 thousand feet per minute. Plan 2: At 150 thousand feet, the module drops at a steady rate of 1 thousand feet per minute. What system of equations represents these plans?

Learn how to model the descent of lunar modules with a system of linear equations. This problem involves calculating the altitude over time with Plan 1 and Plan 2, based on their respective descent rates and initial heights. Suitable for high school students.

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