Questions: Which of the following equations have exactly one solution? Choose all answers that apply: A -6x-6=103x-103 B -6x-6=-6x-103 C 103x-6=103x-103 D -103x-6=-6x-103

Which of the following equations have exactly one solution?
Choose all answers that apply:

A -6x-6=103x-103
B -6x-6=-6x-103
C 103x-6=103x-103

D -103x-6=-6x-103

Transcript text: Which of the following equations have exactly one solution? Choose all answers that apply: A $-6 x-6=103 x-103$ B $-6 x-6=-6 x-103$ (C) $103 x-6=103 x-103$ D $-103 x-6=-6 x-103$

Solution

Solution Steps

To determine which equations have exactly one solution, we need to analyze each equation to see if it simplifies to a form where $ ax = b $ with $ a \neq 0 $. If an equation simplifies to a contradiction or an identity, it does not have exactly one solution.

Step 1: Analyze Equation A

For equation A, $ -6x - 6 = 103x - 103 $, we can rearrange it to find the solution: \[ -6x - 103x = -103 + 6 \implies -109x = -97 \implies x = \frac{97}{109} \] This equation has exactly one solution.

Step 2: Analyze Equation B

For equation B, $ -6x - 6 = -6x - 103 $, simplifying gives: \[ -6 = -103 \] This is a contradiction, indicating that there are no solutions. Thus, it does not have exactly one solution.

Step 3: Analyze Equation C

For equation C, $ 103x - 6 = 103x - 103 $, simplifying leads to: \[ -6 = -103 \] This is also a contradiction, indicating that there are no solutions. Thus, it does not have exactly one solution.

Step 4: Analyze Equation D

For equation D, $ -103x - 6 = -6x - 103 $, we can rearrange it to find the solution: \[ -103x + 6x = -103 + 6 \implies -97x = -97 \implies x = 1 \] This equation has exactly one solution.