Which equations have infinitely many solutions? Select TWO that apply.
A. 6(x - 9) = 6x - 54 + x
B. 5(x - 8) + 10 = 5(x + 2)
C. 20 - 8x - 6 = 2(7 - 4x)
D. 4(x + 4) = 2(2x + 5) + 6
E. 2x| 9 = 18 + 2x
The correct answers are A and C.
A. 6(x - 9) = 6x - 54 + x simplifies to 6x - 54 = 6x - 54 + x. This equation is true for any value of x, so it has infinitely many solutions.
C. 20 - 8x - 6 = 2(7 - 4x) simplifies to 14 - 8x = 14 - 8x. This equation is also true for any value of x, so it has infinitely many solutions.
To determine which equations have infinitely many solutions, we need to look for equations where both sides are equivalent to each other, regardless of the value of the variable.
Let's examine the given options:
A. 6(x - 9) = 6x - 54 + x
Expanding:
6x - 54 = 6x - 54 + x
Combining like terms:
6x - 54 = 7x - 54
Both sides of the equation are equal, regardless of the value of x. Therefore, this equation has infinitely many solutions.
B. 5(x - 8) + 10 = 5(x + 2)
Expanding:
5x - 40 + 10 = 5x + 10
Simplifying:
5x - 30 = 5x + 10
The equation simplifies to -30 = 10, which is not true. Therefore, this equation does not have infinitely many solutions.
C. 20 - 8x - 6 = 2(7 - 4x)
Simplifying:
14 - 8x = 14 - 8x
Both sides of the equation are equal, regardless of the value of x. Therefore, this equation has infinitely many solutions.
D. 4(x + 4) = 2(2x + 5) + 6
Expanding:
4x + 16 = 4x + 10 + 6
Simplifying:
4x + 16 = 4x + 16
Both sides of the equation are equal, regardless of the value of x. Therefore, this equation has infinitely many solutions.
E. 2x | 9 = 18 + 2x
There seems to be an error in this equation. The "|" symbol used here is not recognized as a valid mathematical operator. Please check if there was a typo or clarify the equation.
Based on the analysis, the equations that have infinitely many solutions are:
A. 6(x - 9) = 6x - 54 + x
C. 20 - 8x - 6 = 2(7 - 4x)
D. 4(x + 4) = 2(2x + 5) + 6
To determine which equations have infinitely many solutions, we need to identify the equations that simplify to the same expression on both sides of the equation.
Let's simplify each equation and see which ones meet this condition:
A. 6(x - 9) = 6x - 54 + x
First, distribute the 6 on the left side:
6x - 54 = 6x - 54 + x
Combining like terms, we get:
6x - 54 = 7x - 54
Since the expression on the left side is equal to the expression on the right side, this equation has infinitely many solutions.
B. 5(x - 8) + 10 = 5(x + 2)
First, distribute the 5 on both sides:
5x - 40 + 10 = 5x + 10
Combine like terms:
5x - 30 = 5x + 10
The variables cancel out, resulting in -30 = 10. Since -30 ≠ 10, this equation does not have infinitely many solutions.
C. 20 - 8x - 6 = 2(7 - 4x)
First, simplify the expression on the right side:
20 - 8x - 6 = 14 - 8x
Combine like terms:
14 - 8x = 14 - 8x
Since the expression on the left side is equal to the expression on the right side, this equation has infinitely many solutions.
D. 4(x + 4) = 2(2x + 5) + 6
First, distribute the 4 and 2 on their respective sides:
4x + 16 = 4x + 10 + 6
Combine like terms:
4x + 16 = 4x + 16
Since the expression on the left side is equal to the expression on the right side, this equation has infinitely many solutions.
E. 2x| 9 = 18 + 2x
This equation seems to be missing an operator between 2x and 9. Without a clear equation, it is not possible to determine if it has infinitely many solutions.
Based on our analysis, the equations that have infinitely many solutions are A and C.