To use the substitution method to solve the system of equations \(2x + 3y = 4\) and \(-4x - 6y = -8\), we start by solving one of the equations for one variable and then substituting it into the other.
Let's solve the first equation for \(y\):
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Start with the equation: \[ 2x + 3y = 4 \]
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Solve for \(y\): \[ 3y = 4 - 2x \] \[ y = \frac{4 - 2x}{3} \]
Now we can substitute this expression for \(y\) into the second equation \(-4x - 6y = -8\).
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Substitute \(y\): \[ -4x - 6\left(\frac{4 - 2x}{3}\right) = -8 \]
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Distribute: \[ -4x - \frac{24 - 12x}{3} = -8 \] \[ -4x - 8 + 4x = -8 \]
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Notice that the \( -4x\) and \( +4x\) cancel out: \[ -8 = -8 \]
This is a true statement, which indicates that the second equation is essentially a multiple of the first equation. Both equations represent the same line.
Since both equations are equivalent, we conclude that the system has an infinite number of solutions.
Hence, the correct response would be:
infinite solutions.