Problem 91 - Intermediate Algebra

Question

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{m}{m-2}-\frac{1}{m-3}\) b. \(\frac{m}{m-2}-\frac{1}{m-3}=1\)

Step-by-Step Solution

Answer

a. Simplified expression: \( \frac{m^2 - 4m + 2}{(m-2)(m-3)} \). b. Solution: \( m = 4 \).

Step 1: Identify the Common Denominator (Part a)

To add or subtract fractions such as \( \frac{m}{m-2} \) and \( \frac{1}{m-3} \), we first need to find a common denominator. Here, the common denominator is the product of the two individual denominators: \( (m-2)(m-3) \).

Step 2: Rewrite Each Fraction with Common Denominator (Part a)

Transform each fraction to have the common denominator \( (m-2)(m-3) \).The first fraction: \[ \frac{m}{m-2} = \frac{m(m-3)}{(m-2)(m-3)} \]The second fraction:\[ \frac{1}{m-3} = \frac{1(m-2)}{(m-3)(m-2)} \]

Step 3: Subtract the Fractions (Part a)

Now subtract the rewritten fractions:\[ \frac{m(m-3)}{(m-2)(m-3)} - \frac{(m-2)}{(m-3)(m-2)} \]This results in:\[ \frac{m^2 - 3m - m + 2}{(m-2)(m-3)} = \frac{m^2 - 4m + 2}{(m-2)(m-3)} \]

Step 4: Solve the Equation (Part b)

For \( \frac{m}{m-2} - \frac{1}{m-3} = 1 \), rewrite each side with a common denominator, similar to part (a):\[ \frac{m(m-3) - (m-2)}{(m-2)(m-3)} = 1 \]Simplify the numerator:\[ m^2 - 3m - m + 2 = m^2 - 4m + 2 \]Thus, our equation becomes:\[ \frac{m^2 - 4m + 2}{(m-2)(m-3)} = 1 \]

Step 5: Eliminate Denominator and Solve for m (Part b)

Eliminate the denominator by multiplying both sides of the equation by \((m-2)(m-3)\):\[ m^2 - 4m + 2 = (m-2)(m-3) \]Expanding the right side:\[ m^2 - 4m + 2 = m^2 - 5m + 6 \]Subtract \(m^2\) from both sides:\[ -4m + 2 = -5m + 6 \]Add 5m to both sides:\[ m + 2 = 6 \]Subtract 2 from both sides:\[ m = 4 \]

Step 6: Check the Solution (Part b)

Substitute \(m = 4\) back into the original equation \( \frac{4}{4-2} - \frac{1}{4-3} = 1 \):The left side evaluates to \( \frac{4}{2} - 1 = 2 - 1 = 1 \), which equals the right side confirming our solution.

Key Concepts

Common Denominator

When you come across fractions with different denominators, handling them in calculations means finding a common ground. This is where the concept of the common denominator comes into play. It is a crucial step when adding or subtracting fractions.

Simply put, a common denominator is a shared multiple of the denominators of two or more fractions. For example, if we have two fractions

• \( \frac{m}{m-2} \)
• \( \frac{1}{m-3} \)

to perform operations between them, we need each fraction to have the same denominator. A common method is to multiply the denominators together, which, in this case, results in \((m-2)(m-3)\).

Once we have this unified denominator, we can safely move forward to adjust each fraction accordingly. This step is essential for combining the fractions properly, laying a strong foundation for further simplification or equation solving.

Simplifying Fractions

After finding a common denominator and adjusting the fractions, the next step in working with rational expressions is simplification. Simplifying means breaking down the expression into its simplest form, without changing its value.

In our example, once the common denominator is applied, the expressions become:

• \( \frac{m(m-3)}{(m-2)(m-3)} \)
• \( \frac{1(m-2)}{(m-3)(m-2)} \)

Here, subtraction is performed by uniting the numerators over a shared denominator, giving us:

\[ \frac{m^2 - 3m - m + 2}{(m-2)(m-3)} \]
This simplifies further to \( \frac{m^2 - 4m + 2}{(m-2)(m-3)} \). Such simplification helps in organizing the expression clearly, reducing complexity, and preparing for potentially solving equations or verifying results.

Solving Rational Equations

Solving equations with rational expressions involves manipulating the equation to isolate the variable. Once an equation includes fractions, it's often useful to eliminate denominators by multiplying each part of the equation by the common denominator.

To tackle the example equation:
\[ \frac{m^2 - 4m + 2}{(m-2)(m-3)} = 1 \]
We eliminate the denominator by multiplying each side of the equation by \( (m-2)(m-3) \), leading to:
\[ m^2 - 4m + 2 = m^2 - 5m + 6 \]
This equation is now solvable using straightforward algebraic steps like combining like terms and isolating \( m \).
After simplifying and rearranging, we find \( m = 4 \).
Finally, always verify the solution by plugging it back into the original equation to ensure it satisfies the entire equation, confirming its validity.