Justify why −12(x−3)−52=−16 has one solution, infinitely many solutions, or no solution.

Answer 1

To determine whether the equation −12(x−3)−52=−16 has one solution, infinitely many solutions, or no solution, we need to simplify the equation and see if it leads to a contradiction or if it allows for all values of x to satisfy the equation.

Start by simplifying the left side of the equation:
−12(x−3)−52=−16
Distribute −12 to the terms inside the parentheses:
−12x + 36 - 52 = −16
Combine like terms:
−12x - 16 = −16

Now, we can isolate the variable x by adding 16 to both sides of the equation:
−12x - 16 + 16 = −16 + 16
−12x = 0

To solve for x, divide both sides of the equation by -12:
(-12x) / (-12) = 0 / (-12)
x = 0

Thus, the equation has one solution, which is x = 0.