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Chapter 2: Problem 35

Solve using the addition principle. Don't forget to check! $$ -\frac{1}{5}+z=-\frac{1}{4} $$

Short Answer

Expert verified
The solution is \(z = -\frac{1}{20}\).

Step by step solution

01

- Isolate the Variable

To isolate the variable, add \(-\frac{1}{5}\) to both sides of the equation.\[-\frac{1}{5} + z + \frac{1}{5} = -\frac{1}{4} + \frac{1}{5}\]
02

- Simplify the Equation

Simplify the right-hand side of the equation by finding a common denominator. \[-\frac{1}{4} + \frac{1}{5} = -\frac{5}{20} + \frac{4}{20} = -\frac{1}{20}\] Therefore, the equation becomes \(z = -\frac{1}{20}\).
03

- Verify the Solution

Substitute \(z = -\frac{1}{20}\) back into the original equation to verify: \[-\frac{1}{5} + z = -\frac{1}{4}\] \[-\frac{1}{5} + \left( -\frac{1}{20} \right) = -\frac{4}{20} - \frac{1}{20} = -\frac{5}{20} = -\frac{1}{4}\] Since both sides of the equation are equal, the solution is verified.

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Key Concepts

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

Addition Principle
In algebra, the addition principle helps us solve equations by allowing us to add the same value to both sides. This keeps the equation balanced. For example, in the equation \(-\frac{1}{5} + z = -\frac{1}{4}\), we need to isolate the variable, which means making \(z\) stand alone on one side. By adding \(\frac{1}{5}\) to both sides, we move towards isolating \(z\). This results in: \(-\frac{1}{5} + z + \frac{1}{5} = -\frac{1}{4} + \frac{1}{5}\). Now that \(z\) is by itself on the left side, we can simplify the right side.
Isolating the Variable
Isolating the variable means getting our unknown (usually denoted as \(x\) or \(z\)) alone on one side of the equation. After applying the addition principle, we have: \(-\frac{1}{5} + z + \frac{1}{5} = -\frac{1}{4} + \frac{1}{5}\). Simplifying the left side: \(-\frac{1}{5} + \frac{1}{5}\) cancels out, leaving just \(z\). On the right side, we need to combine the fractions, which leads us to our next concept 鈥� finding a common denominator.
Common Denominator
When dealing with fractions, we often need a common denominator to add or subtract them. In the expression \(-\frac{1}{4} + \frac{1}{5}\), the denominators are 4 and 5. To combine these fractions, find a common multiple of 4 and 5, which is 20. We rewrite the fractions: \(-\frac{1}{4} = -\frac{5}{20}\) and \(\frac{1}{5} = \frac{4}{20}\). Now, we can add them easily: \(-\frac{5}{20} + \frac{4}{20} = -\frac{1}{20}\). This simplifies our problem to \(z = -\frac{1}{20}\).
Verifying Solutions
Always verify your solution by substituting it back into the original equation. We found \(z = -\frac{1}{20}\). Substituting this back, we get: \(-\frac{1}{5} + z = -\frac{1}{4}\). Replace \(z\) with \(-\frac{1}{20}\): \(-\frac{1}{5} + \, (-\frac{1}{20}) = -\frac{1}{4}\). Simplify: \(-\frac{4}{20} - \frac{1}{20} = -\frac{5}{20}= -\frac{1}{4}\). Since both sides match, we've verified the solution accurately!

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