To determine which equation among the given options has the same solution as the given equation [tex]\(-5n + 31 = -14n - 5\)[/tex], we need to solve the original equation and then verify the solutions for each of the provided equations.
Step 1: Solve the original equation
Given:
[tex]\[
-5n + 31 = -14n - 5
\][/tex]
First, we will collect all terms involving [tex]\(n\)[/tex] on one side and constant terms on the other side:
[tex]\[
-5n + 14n = -5 - 31
\][/tex]
Simplifying both sides, we get:
[tex]\[
9n = -36
\][/tex]
Now, solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{-36}{9} = -4
\][/tex]
So, the solution to the equation [tex]\(-5n + 31 = -14n - 5\)[/tex] is [tex]\(n = -4\)[/tex].
Step 2: Verify the solution with each given equation
Equation 1: [tex]\(3 - 6n + 3n = -3 + 4 - 4n\)[/tex]
[tex]\[
3 - 3n = 1 - 4n
\][/tex]
Move all terms involving [tex]\(n\)[/tex] to one side and constants to the other:
[tex]\[
3n - 4n = 1 - 3
\][/tex]
[tex]\[
-n = -2
\][/tex]
[tex]\[
n = 2
\][/tex]
So, the solution for the first equation is [tex]\(n = 2\)[/tex], which does not match our solution.
Equation 2: [tex]\(-1 - 22n = -20n - 9\)[/tex]
[tex]\[
-1 - 22n + 20n = -9
\][/tex]
Simplify and collect like terms:
[tex]\[
-2n = -8
\][/tex]
[tex]\[
n = 4
\][/tex]
So, the solution for the second equation is [tex]\(n = 4\)[/tex], which does not match our solution.
Equation 3: [tex]\(\frac{11}{4}n + 2 = -3 + \frac{3}{2}n\)[/tex]
Multiply through by 4 to clear the fraction:
[tex]\[
11n + 8 = -12 + 6n
\][/tex]
Move all terms involving [tex]\(n\)[/tex] to one side and constants to the other:
[tex]\[
11n - 6n = -12 - 8
\][/tex]
[tex]\[
5n = -20
\][/tex]
[tex]\[
n = -4
\][/tex]
This gives us the same solution, [tex]\(n = -4\)[/tex], matching our original solution.
Equation 4: [tex]\(1 = 0.75n + 3.25\)[/tex]
Subtract 3.25 from both sides:
[tex]\[
1 - 3.25 = 0.75n
\][/tex]
[tex]\[
-2.25 = 0.75n
\][/tex]
Divide both sides by 0.75:
[tex]\[
n = \frac{-2.25}{0.75} = -3
\][/tex]
So, the solution for the fourth equation is [tex]\(n = -3\)[/tex], which does not match our solution.
So, the equation that has the same solution as [tex]\(-5n + 31 = -14n - 5\)[/tex] is:
[tex]\[
\boxed{\frac{11}{4}n + 2 = -3 + \frac{3}{2}n}
\][/tex]
Thus, the correct answer is the third equation.
