We are given the system of equations:

$3x - 4y = 2$ $9x - 12y = 6$

We will solve this using the elimination method.

Step 1: Compare the equations

We can see that the second equation is a multiple of the first one. Specifically, if we multiply the first equation by 3:

$3(3x - 4y) = 3(2)$ This results in: $9x - 12y = 6$

So, the second equation is exactly the same as the first equation multiplied by 3.

Step 2: Interpretation

Since the two equations are not independent (they represent the same line in the $xy$-plane), the system is dependent. This means there are infinitely many solutions, depending on the value of one variable (we can express the solution in terms of a parameter, say $t$).

Step 3: Express the solutions

Let's solve the first equation for $x$ in terms of $y$:

$3x - 4y = 2 \quad \Rightarrow \quad 3x = 4y + 2 \quad \Rightarrow \quad x = \frac{4y + 2}{3}$

Now, express $y$ as a parameter, say $t$. Let $y = t$, so the solution becomes:

$x = \frac{4t + 2}{3}$

Final Solution:

The system has infinitely many solutions, and the solutions can be expressed as:

$x = \frac{4t + 2}{3}, \quad y = t$

where $t$ is a parameter that can take any real value.

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Would you like more details on the steps or have any other questions?

Here are some related questions for you:

1. How would you solve this system using substitution instead of elimination?
2. What happens if the coefficients of $x$ and $y$ in the second equation are different from a multiple of the first equation?
3. How do you recognize dependent systems from their equations?
4. What does it mean for a system of equations to be inconsistent?
5. How do you solve a system of equations with no solution?

Tip: When you see two equations that are multiples of each other, the system is likely dependent and has infinitely many solutions.