At SpaceX, level 1 engineers make a salary of $56,000 a year and level 2 engineers get a yearly salary of $68,000. They have 8 total engineers, all of which are level 1. They are looking to promote some of them to level 2. However, next year they’ll only be able to afford $472,000 in engineer salaries. Write a system of equations befitting this problem where a is the number of level 1 engineers and b is the number of level 2 engineers

We can model this scenario with two equations based on the total number of engineers and the total salary budget.

### Variables:
- Let $ a $ represent the number of level 1 engineers.
- Let $ b $ represent the number of level 2 engineers.

### Information given:
1. There are 8 engineers in total.
2. Level 1 engineers make $56,000 per year.
3. Level 2 engineers make $68,000 per year.
4. The total salary budget is $472,000.

### System of Equations:

1. **Number of engineers**: The total number of engineers is 8.
   \[
   a + b = 8
   \]
   
2. **Salary budget**: The total salary for all engineers is $472,000.
   \[
   56,000a + 68,000b = 472,000
   \]

Thus, the system of equations is:
\[
\begin{aligned}
a + b &= 8 \\
56,000a + 68,000b &= 472,000
\end{aligned}
\]

This system represents the relationship between the number of engineers at each level and their total salary.

Would you like further details on how to solve this system or have any other questions?

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Here are 5 related questions for further exploration:
1. How would you solve this system of equations for $ a $ and $ b $?
2. What happens to the budget if more engineers are promoted to level 2?
3. How would the system change if the salary for level 1 engineers increases next year?
4. What would happen if the total number of engineers changes to 10 instead of 8?
5. How much more will SpaceX need to increase its budget if all engineers are promoted to level 2?

**Tip**: To solve systems of equations, substitution or elimination methods can be useful, depending on which is easier to apply based on the structure of the equations.

At SpaceX, level 1 engineers make a salary of $56,000 a year and level 2 engineers get a yearly salary of $68,000. They have 8 total engineers, all of which are level 1. They are looking to promote some of them to level 2. However, next year they’ll only be able to afford $472,000 in engineer salaries. Write a system of equations befitting this problem where a is the number of level 1 engineers and b is the number of level 2 engineers.

Explore how to set up and solve a system of linear equations to determine the number of level 1 and level 2 engineers SpaceX can afford with a $472,000 salary budget. Understand the algebraic concepts and methods behind solving linear systems. Suitable for high school students in Grades 9-12.

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