To solve the system of equations

3x - 2y = 6 \tag{1} -6x + 4y = -12 \tag{2}

we can use the substitution or elimination method. Here, the elimination method is straightforward due to the coefficients of $x$ and $y$.

Step 1: Simplify the equations if needed

The second equation $-6x + 4y = -12$ can be simplified by dividing through by $-2$: 3x - 2y = 6 \tag{1} 3x - 2y = 6 \tag{2 (simplified)}

Step 2: Observe the relationship between the equations

From the simplification, both equations are identical: $3x - 2y = 6$

This implies that the system of equations is dependent, meaning they represent the same line.

Step 3: Solve for $y$ in terms of $x$

From $3x - 2y = 6$: $-2y = -3x + 6$ $y = \frac{3}{2}x - 3$

Step 4: Describe the solution

The solution to the system of equations is all points $(x, y)$ that satisfy $y = \frac{3}{2}x - 3$.

In conclusion:

• The system has infinitely many solutions.
• The solutions are all points on the line $y = \frac{3}{2}x - 3$.

Let me know if you'd like further explanation or verification using another method!

Related Questions

1. What does it mean when a system of equations is dependent?
2. How can graphing help visualize the solution to this system?
3. How would the solution change if one equation were slightly altered (e.g., $3x - 2y = 7$)?
4. How does elimination work for systems with three variables?
5. What are some real-world applications of dependent systems?

Tip

When solving systems of equations, always check for relationships between equations (e.g., multiples or identical). This can save time and clarify the nature of the solution!