To determine which equations have infinitely many solutions, we need to simplify each equation and see if they result in a true statement or a contradiction.
Let's evaluate each option:
A. \(6(x-9)=6x-54+x\)
Simplifying both sides: Left Side: \(6x - 54\) Right Side: \(6x - 54 + x\) simplifies to \(7x - 54\).
The equation becomes: \(6x - 54 = 7x - 54\)
Subtracting \(6x\) from both sides: \(-54 = x - 54\)
Adding 54 to both sides gives \(x = 0\). This does not have infinitely many solutions.
B. \(5(x - 8) + 10 = 5(x + 2)\)
Simplifying both sides: Left Side: \(5x - 40 + 10 = 5x - 30\) Right Side: \(5x + 10\).
Now the equation becomes \(5x - 30 = 5x + 10\).
Subtracting \(5x\) from both sides yields: \(-30 = 10\), which is a contradiction. Thus, this does not have infinitely many solutions.
C. \(20 - 8x - 6 = 2(7 - 4x)\)
Simplifying both sides: Left Side: \(20 - 6 - 8x = 14 - 8x\). Right Side: This simplifies to \(14 - 8x\).
Thus, the equation becomes: \(14 - 8x = 14 - 8x\).
This is true for all values of \(x\), indicating infinitely many solutions.
D. \(4(x + 4) = 2(2x + 5) + 6\)
Simplifying both sides: Left Side: \(4x + 16\), Right Side: \(4x + 10 + 6 = 4x + 16\).
Thus, the equation becomes: \(4x + 16 = 4x + 16\).
This is true for all values of \(x\), indicating infinitely many solutions.
E. \(2x + 9 = 18 + 2x\)
Subtracting \(2x\) from both sides, we have: \(9 = 18\), which is a contradiction. Thus, this does not have infinitely many solutions.
Based on this analysis, the 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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