To solve the system of equations using the substitution method, we start with the given equations:
- \( 2x + 3y = 4 \) (Equation 1)
- \( -4x - 6y = -8 \) (Equation 2)
First, we notice that Equation 2 is a multiple of Equation 1. In fact, if we multiply Equation 1 by -2, we get:
\[ -2(2x + 3y) = -2(4) \] \[ -4x - 6y = -8 \]
This shows that both equations represent the same line in the coordinate system, meaning they are dependent equations.
Since they represent the same line, there are infinitely many solutions. We can express one variable in terms of the other. Let's solve Equation 1 for \( y \):
From Equation 1: \[ 2x + 3y = 4 \] Subtract \( 2x \) from both sides: \[ 3y = 4 - 2x \] Now, divide by 3: \[ y = \frac{4 - 2x}{3} \]
This gives us the expression for \( y \) in terms of \( x \). Therefore, the solution set can be expressed as: \[ \left( x, \frac{4 - 2x}{3} \right) \] for any real number \( x \).
In conclusion, the system has infinitely many solutions, which can be written as: \[ { (x, \frac{4 - 2x}{3}) | x \in \mathbb{R} } \]