We are given the following system of linear equations:

$- x + 6y = 9$ $- 4x + 24y = 48$

We need to determine if the system has no solutions, infinitely many solutions, or exactly one solution.

Step 1: Compare the equations

Notice that the second equation can be obtained by multiplying the first equation by 4:

$- x + 6y = 9 \quad \text{(first equation)}$ $4(- x + 6y) = 4(9)$ $- 4x + 24y = 36 \quad \text{(this is the modified first equation)}$

Step 2: Analyze the result

Now compare this modified equation to the second original equation:

$- 4x + 24y = 48 \quad \text{(original second equation)}$

Clearly, the two equations are not identical because their constants differ (36 vs. 48). Therefore, these lines are parallel and never intersect. This means there is no solution to the system, as the lines represent parallel lines that do not meet.

Conclusion:

The system of equations has no solutions.

Would you like further details or have any questions?

Related Questions:

1. What is the geometric interpretation of a system with infinitely many solutions?
2. How do you identify if a system has exactly one solution?
3. What conditions lead to a system of equations being inconsistent?
4. Can two linear equations ever represent the same line?
5. How can substitution help in solving a system of equations?

Tip:

If the equations are multiples of each other but have different constants, the system will have no solutions (parallel lines).