What is the value of \(\rm \left(\frac{1}{a}-\frac{1}{b}-\frac{1}{c}\right)\) if, \(\rm \frac{2a-5}{a}-\frac{4b-5}{b}+\frac{6c+5}{c}=0\)?
Solving for an Expression from an Algebraic Equation
The question asks us to find the value of the expression \( \left(\frac{1}{a}-\frac{1}{b}-\frac{1}{c}\right) \) given the algebraic equation \( \frac{2a-5}{a}-\frac{4b-5}{b}+\frac{6c+5}{c}=0 \). To solve this, we need to simplify the given equation and rearrange the terms to match the target expression.
Step-by-Step Solution to Find the Value
Let's break down the given equation term by term. We can rewrite each fraction by dividing the numerator by the denominator:
- The first term is \( \frac{2a-5}{a} \). This can be split as \( \frac{2a}{a} - \frac{5}{a} \). Simplifying \( \frac{2a}{a} \) gives us \( 2 \). So, \( \frac{2a-5}{a} = 2 - \frac{5}{a} \).
- The second term is \( \frac{4b-5}{b} \). This can be split as \( \frac{4b}{b} - \frac{5}{b} \). Simplifying \( \frac{4b}{b} \) gives us \( 4 \). So, \( \frac{4b-5}{b} = 4 - \frac{5}{b} \).
- The third term is \( \frac{6c+5}{c} \). This can be split as \( \frac{6c}{c} + \frac{5}{c} \). Simplifying \( \frac{6c}{c} \) gives us \( 6 \). So, \( \frac{6c+5}{c} = 6 + \frac{5}{c} \).
Now, substitute these simplified terms back into the original equation:
\[ \left(2 - \frac{5}{a}\right) - \left(4 - \frac{5}{b}\right) + \left(6 + \frac{5}{c}\right) = 0 \]
Next, remove the parentheses, being careful with the signs:
\[ 2 - \frac{5}{a} - 4 + \frac{5}{b} + 6 + \frac{5}{c} = 0 \]
Combine the constant terms (2, -4, and 6):
\[ (2 - 4 + 6) + \left(-\frac{5}{a} + \frac{5}{b} + \frac{5}{c}\right) = 0 \]
\[ 4 + \left(-\frac{5}{a} + \frac{5}{b} + \frac{5}{c}\right) = 0 \]
Now, look at the terms involving \( a, b, \) and \( c \) in the parenthesis: \( -\frac{5}{a} + \frac{5}{b} + \frac{5}{c} \). We can factor out \( 5 \) from these terms:
\[ -\frac{5}{a} + \frac{5}{b} + \frac{5}{c} = 5 \left(-\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \]
So the equation becomes:
\[ 4 + 5 \left(-\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) = 0 \]
We are trying to find the value of \( \left(\frac{1}{a}-\frac{1}{b}-\frac{1}{c}\right) \). Notice that the expression inside the parenthesis is \( \left(-\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \), which is the negative of the expression we want to find. That is, \( \left(-\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) = - \left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right) \).
Substitute this into the equation:
\[ 4 + 5 \left(-\left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right)\right) = 0 \]
\[ 4 - 5 \left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right) = 0 \]
Now, isolate the expression \( \left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right) \):
Subtract 4 from both sides:
\[ -5 \left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right) = -4 \]
Divide both sides by -5:
\[ \left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right) = \frac{-4}{-5} \]
\[ \left(\frac{1}{a} - \frac{1}{b} - \frac{1}{c}\right) = \frac{4}{5} \]
Thus, the value of \( \left(\frac{1}{a}-\frac{1}{b}-\frac{1}{c}\right) \) is \( \frac{4}{5} \).
Summary of Key Steps
- Deconstruct each fractional term in the given equation.
- Combine constant terms and terms involving \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \).
- Factor out common coefficients.
- Rearrange the simplified equation to solve for the desired expression.
| Original Equation | Simplified Terms |
|---|---|
| \( \frac{2a-5}{a} \) | \( 2 - \frac{5}{a} \) |
| \( \frac{4b-5}{b} \) | \( 4 - \frac{5}{b} \) |
| \( \frac{6c+5}{c} \) | \( 6 + \frac{5}{c} \) |
Revision Table: Algebraic Simplification
| Concept | Description |
|---|---|
| Separating fractions | A fraction like \( \frac{x+y}{z} \) can be written as \( \frac{x}{z} + \frac{y}{z} \). This was used to simplify each term in the equation. |
| Combining like terms | Constant terms and terms with the same variable structure (like \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \)) are grouped together. |
| Factoring | Extracting a common numerical factor (like 5 in this case) simplifies the expression. |
| Rearranging equations | Using inverse operations (addition/subtraction, multiplication/division) to isolate the variable or expression of interest. |
Additional Information: Working with Algebraic Fractions
When dealing with algebraic fractions like those in this problem, it's often helpful to simplify them first. Expressions in the form \( \frac{Ax+B}{x} \) can always be split into \( \frac{Ax}{x} + \frac{B}{x} = A + \frac{B}{x} \). This technique was crucial in solving the given problem quickly and efficiently.
Understanding how to manipulate fractions and rearrange equations is fundamental in algebra. Always pay close attention to the signs when removing parentheses, especially when a minus sign is in front of the parenthesis.
Practice problems involving simplification of algebraic expressions and solving for specific values or expressions from given equations will help build proficiency.
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