Answer :
Let's analyze each of the given equations to determine which ones have exactly one solution.
### Equation A: [tex]\( 103x - 6 = 103x - 103 \)[/tex]
1. Start by subtracting [tex]\( 103x \)[/tex] from both sides:
[tex]\[
103x - 6 - 103x = 103x - 103 - 103x
\][/tex]
This simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. The statement [tex]\(-6 = -103\)[/tex] is false, indicating that this equation has no solution.
### Equation B: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Start by adding [tex]\( 6x \)[/tex] to both sides:
[tex]\[
-6x - 6 + 6x = -6x - 103 + 6x
\][/tex]
Which simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. Again, the statement [tex]\(-6 = -103\)[/tex] is false, indicating that this equation has no solution.
### Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Attempt to simplify by getting all terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-6x - 103x = -103 + 6
\][/tex]
Simplifies to:
[tex]\[
-109x = -97
\][/tex]
2. Solving for [tex]\( x \)[/tex], divide both sides by [tex]\(-109\)[/tex]:
[tex]\[
x = \frac{97}{109}
\][/tex]
3. This single value for [tex]\( x \)[/tex] means the equation has exactly one solution.
### Equation D: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Attempt to simplify by getting all terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-103x + 6x = -103 + 6
\][/tex]
Simplifies to:
[tex]\[
-97x = -97
\][/tex]
2. Solving for [tex]\( x \)[/tex], divide both sides by [tex]\(-97\)[/tex]:
[tex]\[
x = 1
\][/tex]
3. This single value for [tex]\( x \)[/tex] means the equation has exactly one solution.
In summary, the equations that have exactly one solution are:
- Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
- Equation D: [tex]\(-103x - 6 = -6x - 103\)[/tex]
These equations each result in a single unique value for [tex]\( x \)[/tex].
### Equation A: [tex]\( 103x - 6 = 103x - 103 \)[/tex]
1. Start by subtracting [tex]\( 103x \)[/tex] from both sides:
[tex]\[
103x - 6 - 103x = 103x - 103 - 103x
\][/tex]
This simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. The statement [tex]\(-6 = -103\)[/tex] is false, indicating that this equation has no solution.
### Equation B: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Start by adding [tex]\( 6x \)[/tex] to both sides:
[tex]\[
-6x - 6 + 6x = -6x - 103 + 6x
\][/tex]
Which simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. Again, the statement [tex]\(-6 = -103\)[/tex] is false, indicating that this equation has no solution.
### Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Attempt to simplify by getting all terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-6x - 103x = -103 + 6
\][/tex]
Simplifies to:
[tex]\[
-109x = -97
\][/tex]
2. Solving for [tex]\( x \)[/tex], divide both sides by [tex]\(-109\)[/tex]:
[tex]\[
x = \frac{97}{109}
\][/tex]
3. This single value for [tex]\( x \)[/tex] means the equation has exactly one solution.
### Equation D: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Attempt to simplify by getting all terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-103x + 6x = -103 + 6
\][/tex]
Simplifies to:
[tex]\[
-97x = -97
\][/tex]
2. Solving for [tex]\( x \)[/tex], divide both sides by [tex]\(-97\)[/tex]:
[tex]\[
x = 1
\][/tex]
3. This single value for [tex]\( x \)[/tex] means the equation has exactly one solution.
In summary, the equations that have exactly one solution are:
- Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
- Equation D: [tex]\(-103x - 6 = -6x - 103\)[/tex]
These equations each result in a single unique value for [tex]\( x \)[/tex].
