Which equation has infinitely many solutions?
A. 9x - 3 = 3x + 6× + 2 + 3
B. 12(x+ 8) = 11x - 5
C. 11x - 2x + 15 = 8 + 7 + 9x
D. 5x - 8 = 11 - 7× + 12x
A. 9x - 3 = 3x + 6× + 2 + 3
The equation that has infinitely many solutions is option A: 9x - 3 = 3x + 6× + 2 + 3.
To determine this, we need to identify the equation in which the variables cancel out, resulting in a true statement regardless of the value of x.
Let's simplify each equation to verify which one satisfies this condition.
Option A: 9x - 3 = 3x + 6× + 2 + 3
Combining like terms, we get: 9x - 3 = 9x + 5
Adding 3 to both sides, we have: 9x = 9x + 8
Subtracting 9x from both sides, this simplifies to: 0 = 8
As we can see, the equation ends up as 0 = 8, which is always false. Therefore, option A has no solution, meaning it has infinitely many solutions.
The other options (B, C, and D) do not have this property and will result in a specific value of x when simplified.
To determine which equation has infinitely many solutions, we can simplify each equation and check if the variables cancel out or if we end up with a true statement.
Let's go through each equation one by one:
A. 9x - 3 = 3x + 6× + 2 + 3
Simplifying the equation, we get:
9x - 3 = 9x + 5
Subtracting 9x from both sides, we get:
-3 = 5
This is not a true statement, so there is no solution for this equation.
B. 12(x+ 8) = 11x - 5
Expanding the equation, we get:
12x + 96 = 11x - 5
Subtracting 11x from both sides, we get:
x + 96 = -5
Subtracting 96 from both sides, we get:
x = -101
This is a specific solution, so it does not have infinitely many solutions.
C. 11x - 2x + 15 = 8 + 7 + 9x
Combining like terms, we get:
9x + 15 = 15 + 9x
Subtracting 9x from both sides, we get:
15 = 15
This is a true statement. The variables cancel out, and the equation is satisfied no matter what value x is. Therefore, this equation has infinitely many solutions.
D. 5x - 8 = 11 - 7× + 12x
Combining like terms, we get:
5x - 8 = -7 + 12x
Subtracting 5x from both sides, we get:
-8 = -7x
Dividing both sides by -7, we get:
x = 8/7
This is a specific solution, so it does not have infinitely many solutions.
In conclusion, the equation with infinitely many solutions is option C: 11x - 2x + 15 = 8 + 7 + 9x.