Answer :
Let's go through each equation and determine which ones have exactly one solution.
Equation A: [tex]\( 103x - 6 = 103x - 103 \)[/tex]
To solve this, subtract [tex]\( 103x \)[/tex] from both sides:
[tex]\[ 103x - 103x - 6 = 103x - 103x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = -103 \][/tex]
This is not true, so there is no solution for this equation.
Equation B: [tex]\( -6x - 6 = -6x - 103 \)[/tex]
Subtract [tex]\(-6x\)[/tex] from both sides:
[tex]\[ -6x + 6x - 6 = -6x + 6x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = -103 \][/tex]
This is also not true, so there is no solution for this equation.
Equation C: [tex]\( -103x - 6 = -6x - 103 \)[/tex]
Let's move all terms involving [tex]\( x \)[/tex] to one side:
Add [tex]\( 103x \)[/tex] to both sides:
[tex]\[ -103x + 103x - 6 = -6x + 103x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = 97x - 103 \][/tex]
Add 103 to both sides:
[tex]\[ -6 + 103 = 97x \][/tex]
[tex]\[ 97 = 97x \][/tex]
Divide both sides by 97:
[tex]\[ x = 1 \][/tex]
This gives exactly one solution, [tex]\( x = 1 \)[/tex].
Equation D: [tex]\( -6x - 6 = 103x - 103 \)[/tex]
Let's move all terms involving [tex]\( x \)[/tex] to one side:
Add [tex]\( 6x \)[/tex] to both sides:
[tex]\[ -6x + 6x - 6 = 103x + 6x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = 109x - 103 \][/tex]
Add 103 to both sides:
[tex]\[ -6 + 103 = 109x \][/tex]
[tex]\[ 97 = 109x \][/tex]
Divide both sides by 109:
[tex]\[ x = \frac{97}{109} \][/tex]
This gives exactly one solution, [tex]\( x = \frac{97}{109} \)[/tex].
Therefore, the equations that have exactly one solution are C and D.
Equation A: [tex]\( 103x - 6 = 103x - 103 \)[/tex]
To solve this, subtract [tex]\( 103x \)[/tex] from both sides:
[tex]\[ 103x - 103x - 6 = 103x - 103x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = -103 \][/tex]
This is not true, so there is no solution for this equation.
Equation B: [tex]\( -6x - 6 = -6x - 103 \)[/tex]
Subtract [tex]\(-6x\)[/tex] from both sides:
[tex]\[ -6x + 6x - 6 = -6x + 6x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = -103 \][/tex]
This is also not true, so there is no solution for this equation.
Equation C: [tex]\( -103x - 6 = -6x - 103 \)[/tex]
Let's move all terms involving [tex]\( x \)[/tex] to one side:
Add [tex]\( 103x \)[/tex] to both sides:
[tex]\[ -103x + 103x - 6 = -6x + 103x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = 97x - 103 \][/tex]
Add 103 to both sides:
[tex]\[ -6 + 103 = 97x \][/tex]
[tex]\[ 97 = 97x \][/tex]
Divide both sides by 97:
[tex]\[ x = 1 \][/tex]
This gives exactly one solution, [tex]\( x = 1 \)[/tex].
Equation D: [tex]\( -6x - 6 = 103x - 103 \)[/tex]
Let's move all terms involving [tex]\( x \)[/tex] to one side:
Add [tex]\( 6x \)[/tex] to both sides:
[tex]\[ -6x + 6x - 6 = 103x + 6x - 103 \][/tex]
This simplifies to:
[tex]\[ -6 = 109x - 103 \][/tex]
Add 103 to both sides:
[tex]\[ -6 + 103 = 109x \][/tex]
[tex]\[ 97 = 109x \][/tex]
Divide both sides by 109:
[tex]\[ x = \frac{97}{109} \][/tex]
This gives exactly one solution, [tex]\( x = \frac{97}{109} \)[/tex].
Therefore, the equations that have exactly one solution are C and D.
