ustify why −12(x−3)−52=−16

Question

has one solution, infinitely many solutions, or no solution.(1 point)
Responses

It has infinitely many solutions because the solution is in the form a=a
.
It has infinitely many solutions because the solution is in the form a equals a .

It has one solution because the solution is in the form x=a
. The solution is x=0
.
It has one solution because the solution is in the form x equals a . The solution is x equals 0 .

It has no solution because the solution is in the form a=b
.
It has no solution because the solution is in the form a equals b .

It has one solution, because the solution is in the form x=a
. The solution is x=−12
.

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 it.

1. Distribute \(-12\):

   \[ -12(x - 3) = -12x + 36 \] So the equation becomes: \[ -12x + 36 - 52 = -16 \]

2. Combine like terms:

   \[ -12x + 36 - 52 = -16 \implies -12x - 16 = -16 \]

3. Next, add 16 to both sides of the equation:

   \[ -12x - 16 + 16 = -16 + 16 \implies -12x = 0 \]

4. Now, divide both sides by -12:

   \[ x = 0 \]

Since we arrived at a specific value for \(x\) (which is \(x = 0\)), this indicates that the equation has one solution.

Thus, the correct justification is: It has one solution because the solution is in the form \(x = a\). The solution is \(x = 0\).