To determine which equations have infinitely many solutions, we need to simplify each equation and check for conditions that lead to an identity (when both sides of the equation are identical for all values of the variable).
Option A: \[ 6(x - 9) = 6x - 54 + x \] Expanding both sides: \[ 6x - 54 = 6x - 54 + x \] Combining like terms on the right: \[ 6x - 54 = 7x - 54 \] Subtracting \(6x - 54\) from both sides gives: \[ 0 = x \] This is not an identity, so this equation does not have infinitely many solutions.
Option B: \[ 5(x - 8) + 10 = 5(x + 2) \] Expanding both sides: \[ 5x - 40 + 10 = 5x + 10 \] This simplifies to: \[ 5x - 30 = 5x + 10 \] Subtracting \(5x\) from both sides gives: \[ -30 = 10 \] This is a contradiction, so this equation does not have any solutions.
Option C: \[ 20 - 8x - 6 = 2(7 - 4x) \] Simplifying both sides: \[ 14 - 8x = 14 - 8x \] Since both sides are identical, this is an identity. Thus, this equation has infinitely many solutions.
Option D: \[ 4(x + 4) = 2(2x + 5) + 6 \] Expanding both sides: \[ 4x + 16 = 4x + 10 + 6 \] This simplifies to: \[ 4x + 16 = 4x + 16 \] Since both sides are identical, this is another identity. Thus, this equation also has infinitely many solutions.
The two equations that have infinitely many solutions are:
C. 20 - 8x - 6 = 2(7 - 4x)
D. 4(x + 4) = 2(2x + 5) + 6
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