To solve the system of equations using the substitution method, we first have the two equations:
- \( 2x + 3y = 4 \)
- \( -4x - 6y = -8 \)
Notice that the second equation can be simplified. Divide every term in the second equation by -2:
\[ -4x - 6y = -8 \implies 2x + 3y = 4 \]
Now our system looks like this:
- \( 2x + 3y = 4 \)
- \( 2x + 3y = 4 \)
Both equations are actually the same line. This means the system has infinitely many solutions along the line represented by the equation \( 2x + 3y = 4 \).
To express the solutions parametrically, we can solve for \( y \) in terms of \( x \):
\[ 3y = 4 - 2x \] \[ y = \frac{4 - 2x}{3} \]
Thus, we have:
\[ \text{Solutions: } (x, y) = \left( x, \frac{4 - 2x}{3} \right) \quad \text{for any real number } x. \]
In conclusion, the solution set to the system of equations is infinite, represented as:
\[ { (x, \frac{4 - 2x}{3}) : x \in \mathbb{R} } \]