To use the substitution method to solve the system of equations given by:
- \( 2x + 3y = 4 \)
- \( -4x - 6y = -8 \)
First, we can solve one of the equations for one variable in terms of the other. Let's solve the first equation for \( y \):
\[ 2x + 3y = 4 \] \[ 3y = 4 - 2x \] \[ y = \frac{4 - 2x}{3} \]
Now we can substitute this expression for \( y \) into the second equation:
\[ -4x - 6y = -8 \]
Substituting \( y \):
\[ -4x - 6\left(\frac{4 - 2x}{3}\right) = -8 \]
Multiplying through by 3 to eliminate the fraction:
\[ -12x - 6(4 - 2x) = -24 \] \[ -12x - 24 + 12x = -24 \]
Notice that \( -12x + 12x = 0 \), simplifying gives us:
\[ -24 = -24 \]
This statement is always true, meaning that the two equations are essentially the same. Thus, there are infinitely many solutions that satisfy both equations.
In conclusion, the correct response is:
infinite solutions.