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Answer :
To determine which equations have exactly one solution, let's analyze each of the given equations:
(A) [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\[
-6 = -103
\][/tex]
This results in a false statement, which means this equation has no solution.
(B) [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Add [tex]\(6x\)[/tex] to both sides:
[tex]\[
-6 = -103
\][/tex]
This also gives a false statement, indicating no solution for this equation.
(C) [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Add [tex]\(103x\)[/tex] and [tex]\(103\)[/tex] to both sides:
[tex]\[
-6 + 103 = 103x - 6x
\][/tex]
2. Simplify both sides:
[tex]\[
97 = 97x
\][/tex]
3. Divide both sides by 97:
[tex]\[
x = 1
\][/tex]
This equation has exactly one solution.
(D) [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Add [tex]\(6x\)[/tex] and [tex]\(103\)[/tex] to both sides:
[tex]\[
-6 + 103 = 103x + 6x
\][/tex]
2. Simplify both sides:
[tex]\[
97 = 109x
\][/tex]
3. Divide both sides by 109:
[tex]\[
x = \frac{97}{109}
\][/tex]
This equation also has exactly one solution.
In conclusion, equations (C) and (D) have exactly one solution each.
(A) [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\[
-6 = -103
\][/tex]
This results in a false statement, which means this equation has no solution.
(B) [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Add [tex]\(6x\)[/tex] to both sides:
[tex]\[
-6 = -103
\][/tex]
This also gives a false statement, indicating no solution for this equation.
(C) [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Add [tex]\(103x\)[/tex] and [tex]\(103\)[/tex] to both sides:
[tex]\[
-6 + 103 = 103x - 6x
\][/tex]
2. Simplify both sides:
[tex]\[
97 = 97x
\][/tex]
3. Divide both sides by 97:
[tex]\[
x = 1
\][/tex]
This equation has exactly one solution.
(D) [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Add [tex]\(6x\)[/tex] and [tex]\(103\)[/tex] to both sides:
[tex]\[
-6 + 103 = 103x + 6x
\][/tex]
2. Simplify both sides:
[tex]\[
97 = 109x
\][/tex]
3. Divide both sides by 109:
[tex]\[
x = \frac{97}{109}
\][/tex]
This equation also has exactly one solution.
In conclusion, equations (C) and (D) have exactly one solution each.
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