To determine which equations have infinitely many solutions, we need to simplify and analyze each equation. Equations with infinitely many solutions typically become identities (true for all values of the variable).
Let's analyze each option:
A. \( 6(x - 9) = 6x - 54 + x \)
Simplifying both sides:
- Left side: \( 6(x - 9) = 6x - 54 \)
- Right side: \( 6x - 54 + x = 7x - 54 \)
Setting both sides equal: \( 6x - 54 = 7x - 54 \)
Subtracting \( 6x \) from both sides: \( -54 = x - 54 \)
Adding \( 54 \) to both sides: \( 0 = x \) (This has exactly one solution.)
B. \( 5(x - 8) + 10 = 5(x + 2) \)
Simplifying both sides:
- Left side: \( 5(x - 8) + 10 = 5x - 40 + 10 = 5x - 30 \)
- Right side: \( 5(x + 2) = 5x + 10 \)
Setting both sides equal: \( 5x - 30 = 5x + 10 \)
Subtracting \( 5x \) from both sides: \( -30 = 10 \) (This is a contradiction, so it has no solution.)
C. \( 20 - 8x - 6 = 2(7 - 4x) \)
Simplifying both sides:
- Left side: \( 20 - 8x - 6 = 14 - 8x \)
- Right side: \( 2(7 - 4x) = 14 - 8x \)
Setting both sides equal: \( 14 - 8x = 14 - 8x \) (This is an identity, so it has infinitely many solutions.)
D. \( 4(x + 4) = 2(2x + 5) + 6 \)
Simplifying both sides:
- Left side: \( 4(x + 4) = 4x + 16 \)
- Right side: \( 2(2x + 5) + 6 = 4x + 10 + 6 = 4x + 16 \)
Setting both sides equal: \( 4x + 16 = 4x + 16 \) (This is also an identity, so it has infinitely many solutions.)
In summary, the equations with infinitely many solutions are:
C. \( 20 - 8x - 6 = 2(7 - 4x) \)
D. \( 4(x + 4) = 2(2x + 5) + 6 \)
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