Get the answers you've been searching for with IDNLearn.com. Find reliable solutions to your questions quickly and easily with help from our experienced experts.
Sagot :
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.
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.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.
