Answer :
To determine which of the given equations have exactly one solution, we need to analyze each equation individually:
Equation A: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Simplify both sides of the equation. We can subtract [tex]\(-6x\)[/tex] from both sides:
[tex]\[
-6x - 6 + 6x = -6x - 103 + 6x
\][/tex]
This simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. The equation [tex]\(-6 = -103\)[/tex] is not true, so this equation does not have any solutions.
Equation B: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Simplify both sides of the equation. Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\[
103x - 6 - 103x = 103x - 103 - 103x
\][/tex]
This simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. Again, the equation [tex]\(-6 = -103\)[/tex] is not true, so this equation does not have any solutions.
Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side, and constant terms to the other side:
[tex]\[
-6x - 6 = 103x - 103
\][/tex]
Add [tex]\(6x\)[/tex] to both sides:
[tex]\[
-6 = 103x + 6x - 103
\][/tex]
Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
-6 = 109x - 103
\][/tex]
Add [tex]\(103\)[/tex] to both sides:
[tex]\[
97 = 109x
\][/tex]
Divide both sides by [tex]\(109\)[/tex]:
[tex]\[
x = \frac{97}{109}
\][/tex]
2. This gives us one solution for this equation.
Equation D: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Rearrange the terms to get all [tex]\(x\)[/tex] terms on one side and constants on the other side:
[tex]\[
-103x - 6 = -6x - 103
\][/tex]
Add [tex]\(103x\)[/tex] to both sides:
[tex]\[
-6 = 97x - 103
\][/tex]
Add [tex]\(103\)[/tex] to both sides:
[tex]\[
97 = 97x
\][/tex]
Divide both sides by [tex]\(97\)[/tex]:
[tex]\[
x = 1
\][/tex]
2. This gives us one solution for this equation.
Based on these analyses, Equations C and D each have exactly one solution. Therefore, the correct choices are C and D.
Equation A: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Simplify both sides of the equation. We can subtract [tex]\(-6x\)[/tex] from both sides:
[tex]\[
-6x - 6 + 6x = -6x - 103 + 6x
\][/tex]
This simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. The equation [tex]\(-6 = -103\)[/tex] is not true, so this equation does not have any solutions.
Equation B: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Simplify both sides of the equation. Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\[
103x - 6 - 103x = 103x - 103 - 103x
\][/tex]
This simplifies to:
[tex]\[
-6 = -103
\][/tex]
2. Again, the equation [tex]\(-6 = -103\)[/tex] is not true, so this equation does not have any solutions.
Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side, and constant terms to the other side:
[tex]\[
-6x - 6 = 103x - 103
\][/tex]
Add [tex]\(6x\)[/tex] to both sides:
[tex]\[
-6 = 103x + 6x - 103
\][/tex]
Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
-6 = 109x - 103
\][/tex]
Add [tex]\(103\)[/tex] to both sides:
[tex]\[
97 = 109x
\][/tex]
Divide both sides by [tex]\(109\)[/tex]:
[tex]\[
x = \frac{97}{109}
\][/tex]
2. This gives us one solution for this equation.
Equation D: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Rearrange the terms to get all [tex]\(x\)[/tex] terms on one side and constants on the other side:
[tex]\[
-103x - 6 = -6x - 103
\][/tex]
Add [tex]\(103x\)[/tex] to both sides:
[tex]\[
-6 = 97x - 103
\][/tex]
Add [tex]\(103\)[/tex] to both sides:
[tex]\[
97 = 97x
\][/tex]
Divide both sides by [tex]\(97\)[/tex]:
[tex]\[
x = 1
\][/tex]
2. This gives us one solution for this equation.
Based on these analyses, Equations C and D each have exactly one solution. Therefore, the correct choices are C and D.
