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Chapter 2: Problem 115

Solve each equation requiring simplification. $$\frac{7}{8} n-\frac{3}{4} n=9+2$$

Short Answer

Expert verified
The solution is \(n = 88\).

Step by step solution

01

Combine Like Terms

To solve the equation \(\frac{7}{8} n-\frac{3}{4} n=9+2\), first combine the like terms on the left side. \(\frac{3}{4}\) can be written as \(\frac{6}{8}\), so the equation becomes \(\frac{7}{8} n - \frac{6}{8} n = 9 + 2\). Simplifying the left side, we get \(\frac{1}{8} n = 11\).
02

Isolate the Variable

Next, to isolate \(n\), multiply both sides of the equation by 8, the reciprocal of \(\frac{1}{8}\): \(n = 11 \times 8\).
03

Simplify the Expression

Finally, multiply to find the value of \(n\). \(n = 88\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
When solving linear equations, the first step often involves combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, we have the equation \(\frac{7}{8} n - \frac{3}{4} n = 9 + 2\). To combine these terms, they must be expressed with a common denominator. Here, \(\frac{3}{4}\) is rewritten as \(\frac{6}{8}\). Now we combine: \(\frac{7}{8} n - \frac{6}{8} n\) which simplifies to \(\frac{1}{8} n\). Successfully combining like terms sets the stage for solving for the variable.
Isolating the Variable
After combining like terms, the next step in solving an equation is isolating the variable. In our example, the equation has been simplified to \(\frac{1}{8} n = 11\). To isolate \(n\), we need to eliminate the fraction. This can be done by multiplying both sides of the equation by the reciprocal of \(\frac{1}{8}\), which is 8. Therefore, \( n = 11 \times 8\). Isolating the variable means rewriting the equation so that we have \(n\) on one side and a number on the other.
Simplification in Algebra
The final step in the process is simplification. This means performing any remaining arithmetic to find the value of the variable. From our previous step, we have \(n = 11 \times 8\). When we multiply these numbers, we get \(n = 88\). Simplification in algebra involves ensuring the equation is fully resolved, with the variable clearly isolated and all arithmetic operations performed. By following these systematic steps, you can solve even complex linear equations with ease.

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