Solve the Linear System: 3x - 4y = 10 and 9x - 12y = 15

To solve this problem, we'll follow these steps:

• Step 1: Verify if both equations in the system are proportional.
• Step 2: Determine if they result in a true or false statement when simplified.
• Step 3: Conclude about the solution set based on the results.

Now, let's work through each step:
Step 1: Start with the system of equations:
$\begin{cases} 3x - 4y = 10 \\ 9x - 12y = 15 \end{cases}$

Step 2: Check if the second equation is a multiple of the first equation.
Divide each coefficient of the second equation by 3:
- $(9x) \div 3 = 3x$
- $(-12y) \div 3 = -4y$
- $(15) \div 3 = 5$
Thus, converting: $\begin{cases} 3x - 4y = 10 \\ 3x - 4y = 5 \end{cases}$

Step 3: We notice that while the left sides of both equations are identical, the right sides differ:
This results in a logical contradiction because $10 \neq 5$.
Thus, these lines are parallel and distinct, indicating that the system has no common points of intersection, hence no solution.

Therefore, the correct conclusion for this system of equations is No solution.