High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Which of the following equations have exactly one solution? Choose all answers that apply:

A. [tex]-6x - 6 = -6x - 103[/tex]

B. [tex]-103x - 6 = -6x - 103[/tex]

C. [tex]103x - 6 = 103x - 103[/tex]

D. [tex]-6x - 6 = 103x - 103[/tex]

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].