To determine which equations have infinitely many solutions, we should look for equations that simplify to a true statement (like \(0 = 0\)) after simplifying both sides. Let's analyze each option:
A. \( 6 ( x - 9 ) = 6x - 54 + x \)
Simplifying the left side: \[ 6x - 54 \]
The right side: \[ 6x - 54 + x = 7x - 54 \]
So, the equation is: \[ 6x - 54 = 7x - 54 \]
Subtract \(6x\) from both sides: \[ -54 = x - 54 \]
Adding \(54\) to both sides: \[ 0 = x \] This results in a single solution \( x = 0 \).
B. \( 5 ( x - 8 ) + 10 = 5 ( x + 2 ) \)
Left side: \[ 5x - 40 + 10 = 5x - 30 \]
Right side: \[ 5x + 10 \]
Setting them equal: \[ 5x - 30 = 5x + 10 \]
Subtract \(5x\) from both sides: \[ -30 = 10 \] This is a false statement, therefore no solutions exist.
C. \( 20 - 8x - 6 = 2 ( 7 - 4x ) \)
Left side: \[ 20 - 6 - 8x = 14 - 8x \]
Right side: \[ 14 - 8x \]
Setting them equal: \[ 14 - 8x = 14 - 8x \]
This is a true statement for all values of \(x\), meaning there are infinitely many solutions.
D. \( 4 ( x + 4 ) = 2 ( 2x + 5 ) + 6 \)
Left side: \[ 4x + 16 \]
Right side: \[ 4x + 10 + 6 = 4x + 16 \]
Setting them equal: \[ 4x + 16 = 4x + 16 \]
This is also a true statement for all values of \(x\), meaning there are infinitely many solutions.
E. \( 2x + 9 = 18 + 2x \)
Subtract \(2x\) from both sides: \[ 9 = 18 \] This is a false statement, therefore no solutions exist.
Based on the analysis, the equations that have infinitely many solutions are:
C. \(20 - 8x - 6 = 2 (7 - 4x)\)
D. \(4 (x + 4) = 2 (2x + 5) + 6\)