Answer :
Sure! Let's solve each equation to determine which ones have exactly one solution:
(A) [tex]\(-6x - 6 = -6x - 103\)[/tex]
Start by simplifying both sides of the equation. You'll notice that the [tex]\(-6x\)[/tex] terms on both sides cancel each other out:
[tex]\[-6 = -103\][/tex]
This statement is false because [tex]\(-6\)[/tex] does not equal [tex]\(-103\)[/tex]. Therefore, this equation has no solutions.
(B) [tex]\(-103x - 6 = -6x - 103\)[/tex]
First, let's move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[-103x + 6x = -103 + 6\][/tex]
Combine like terms:
[tex]\[-97x = -97\][/tex]
Next, solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-97\)[/tex]:
[tex]\[x = \frac{-97}{-97}\][/tex]
[tex]\[x = 1\][/tex]
This equation has exactly one solution, [tex]\(x = 1\)[/tex].
(C) [tex]\(103x - 6 = 103x - 103\)[/tex]
Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\[-6 = -103\][/tex]
This statement is also false, similar to equation (A), since [tex]\(-6\)[/tex] does not equal [tex]\(-103\)[/tex]. Hence, this equation has no solutions.
(D) [tex]\(-6x - 6 = 103x - 103\)[/tex]
Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[-6x - 103x = -103 + 6\][/tex]
Combine like terms:
[tex]\[-109x = -97\][/tex]
Now, solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-109\)[/tex]:
[tex]\[x = \frac{-97}{-109}\][/tex]
[tex]\[x = \frac{97}{109}\][/tex]
This equation has exactly one solution, [tex]\(x = \frac{97}{109}\)[/tex].
In summary, the equations that have exactly one solution are (B) and (D). Equation (B) has the solution [tex]\(x = 1\)[/tex], and equation (D) has the solution [tex]\(x = \frac{97}{109}\)[/tex].
(A) [tex]\(-6x - 6 = -6x - 103\)[/tex]
Start by simplifying both sides of the equation. You'll notice that the [tex]\(-6x\)[/tex] terms on both sides cancel each other out:
[tex]\[-6 = -103\][/tex]
This statement is false because [tex]\(-6\)[/tex] does not equal [tex]\(-103\)[/tex]. Therefore, this equation has no solutions.
(B) [tex]\(-103x - 6 = -6x - 103\)[/tex]
First, let's move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[-103x + 6x = -103 + 6\][/tex]
Combine like terms:
[tex]\[-97x = -97\][/tex]
Next, solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-97\)[/tex]:
[tex]\[x = \frac{-97}{-97}\][/tex]
[tex]\[x = 1\][/tex]
This equation has exactly one solution, [tex]\(x = 1\)[/tex].
(C) [tex]\(103x - 6 = 103x - 103\)[/tex]
Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\[-6 = -103\][/tex]
This statement is also false, similar to equation (A), since [tex]\(-6\)[/tex] does not equal [tex]\(-103\)[/tex]. Hence, this equation has no solutions.
(D) [tex]\(-6x - 6 = 103x - 103\)[/tex]
Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[-6x - 103x = -103 + 6\][/tex]
Combine like terms:
[tex]\[-109x = -97\][/tex]
Now, solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-109\)[/tex]:
[tex]\[x = \frac{-97}{-109}\][/tex]
[tex]\[x = \frac{97}{109}\][/tex]
This equation has exactly one solution, [tex]\(x = \frac{97}{109}\)[/tex].
In summary, the equations that have exactly one solution are (B) and (D). Equation (B) has the solution [tex]\(x = 1\)[/tex], and equation (D) has the solution [tex]\(x = \frac{97}{109}\)[/tex].
