QuestionSolve for $n$ in the equation $-\frac{783}{8}=\frac{1}{4} n+\frac{13}{8}\left(\frac{15}{2} n+1\right)$ .

Studdy Solution

STEP 1

Assumptions1. The equation is $-\frac{783}{8}=\frac{1}{4} n+\frac{13}{8}\left(\frac{15}{} n+1\right)$ . We are asked to solve for $n$ .

STEP 2

First, we need to simplify the right-hand side of the equation. We can do this by distributing the $\frac{13}{8}$ to both terms inside the parentheses.
$-\frac{783}{8} = \frac{1}{4} n + \frac{13}{8} \cdot \frac{15}{2} n + \frac{13}{8} \cdot1$

STEP 3

Now, simplify the multiplication of fractions by multiplying the numerators together and the denominators together.
$-\frac{783}{8} = \frac{1}{} n + \frac{13 \cdot15}{8 \cdot2} n + \frac{13}{8}$

STEP 4

implify the fractions.
$-\frac{783}{8} = \frac{1}{4} n + \frac{195}{16} n + \frac{13}{8}$

STEP 5

To simplify further, we need to have the same denominator for all terms. Convert the $\frac{1}{4}n$ term to have a denominator of16.
$-\frac{783}{8} = \frac{4}{16} n + \frac{195}{16} n + \frac{13}{8}$

STEP 6

Now, combine like terms on the right-hand side of the equation.
$-\frac{783}{8} = \frac{199}{16} n + \frac{13}{8}$

STEP 7

To isolate $n$ , we need to move $\frac{13}{}$ to the left-hand side of the equation. We can do this by subtracting $\frac{13}{}$ from both sides of the equation.
$-\frac{783}{} - \frac{13}{} = \frac{199}{16} n$

STEP 8

implify the left-hand side of the equation.
$-\frac{796}{8} = \frac{199}{16} n$

STEP 9

implify the fraction on the left-hand side of the equation.
$-99 = \frac{199}{16} n$

STEP 10

Finally, to solve for $n$ , we need to divide both sides of the equation by $\frac{199}{16}$ .
$n = \frac{-99}{\frac{199}{16}}$

STEP 11

implify the right-hand side of the equation by multiplying $-99$ by the reciprocal of $\frac{199}{16}$ .
$n = -99 \cdot \frac{16}{199}$

STEP 12

implify the multiplication to find the value of $n$ .
$n = -\frac{1584}{199}$ So, $n = -\frac{1584}{199}$ .

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