Answer :
To determine which equations have exactly one solution, let's analyze each given equation separately.
Equation (A): [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side of the equation:
[tex]\(-103x + 6x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(-97x = -97\)[/tex]
3. Divide both sides by [tex]\(-97\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\(x = 1\)[/tex]
This equation has exactly one solution: [tex]\(x = 1\)[/tex].
Equation (B): [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(-6x + 6x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(0 = -97\)[/tex]
Since this statement is false, this equation has no solution.
Equation (C): [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(-6x - 103x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(-109x = -97\)[/tex]
3. Divide both sides by [tex]\(-109\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\(x = \frac{97}{109}\)[/tex]
This equation has exactly one solution: [tex]\(x = \frac{97}{109}\)[/tex].
Equation (D): [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(103x - 103x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(0 = -97\)[/tex]
Since this statement is false, this equation has no solution.
Thus, the equations that have exactly one solution are:
- (A) [tex]\(-103x - 6 = -6x - 103\)[/tex]
- (C) [tex]\(-6x - 6 = 103x - 103\)[/tex]
So, the correct answers are (A) and (C).
Equation (A): [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side of the equation:
[tex]\(-103x + 6x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(-97x = -97\)[/tex]
3. Divide both sides by [tex]\(-97\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\(x = 1\)[/tex]
This equation has exactly one solution: [tex]\(x = 1\)[/tex].
Equation (B): [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(-6x + 6x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(0 = -97\)[/tex]
Since this statement is false, this equation has no solution.
Equation (C): [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(-6x - 103x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(-109x = -97\)[/tex]
3. Divide both sides by [tex]\(-109\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\(x = \frac{97}{109}\)[/tex]
This equation has exactly one solution: [tex]\(x = \frac{97}{109}\)[/tex].
Equation (D): [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\(103x - 103x = -103 + 6\)[/tex]
2. Simplify it:
[tex]\(0 = -97\)[/tex]
Since this statement is false, this equation has no solution.
Thus, the equations that have exactly one solution are:
- (A) [tex]\(-103x - 6 = -6x - 103\)[/tex]
- (C) [tex]\(-6x - 6 = 103x - 103\)[/tex]
So, the correct answers are (A) and (C).
