Express $\frac{2}{3 m-2}-\frac{1}{2 m-3}$ as a single fraction in its lowest terms.
Answer (2)
Find a common denominator: ( 3 m − 2 ) ( 2 m − 3 ) .
Rewrite the fractions with the common denominator and combine them: ( 3 m − 2 ) ( 2 m − 3 ) 2 ( 2 m − 3 ) − ( 3 m − 2 ) .
Simplify the numerator: 2 ( 2 m − 3 ) − ( 3 m − 2 ) = m − 4 .
Expand the denominator and write the final expression: 6 m 2 − 13 m + 6 m − 4 .
Explanation
Problem Analysis We are given the expression 3 m − 2 2 − 2 m − 3 1 . Our goal is to combine these two fractions into a single fraction in its simplest form.
Finding Common Denominator To combine the fractions, we need to find a common denominator. The common denominator is the product of the two denominators, which is ( 3 m − 2 ) ( 2 m − 3 ) .
Rewriting Fractions Now, we rewrite each fraction with the common denominator: 3 m − 2 2 − 2 m − 3 1 = ( 3 m − 2 ) ( 2 m − 3 ) 2 ( 2 m − 3 ) − ( 3 m − 2 ) ( 2 m − 3 ) 1 ( 3 m − 2 )
Combining Fractions Next, we combine the fractions by subtracting the numerators: ( 3 m − 2 ) ( 2 m − 3 ) 2 ( 2 m − 3 ) − ( 3 m − 2 )
Simplifying Numerator Now, we simplify the numerator: 2 ( 2 m − 3 ) − ( 3 m − 2 ) = 4 m − 6 − 3 m + 2 = m − 4
Simplified Expression So, the expression becomes: ( 3 m − 2 ) ( 2 m − 3 ) m − 4
Expanding Denominator and Final Simplification Finally, we expand the denominator: ( 3 m − 2 ) ( 2 m − 3 ) = 3 m ( 2 m ) + 3 m ( − 3 ) − 2 ( 2 m ) − 2 ( − 3 ) = 6 m 2 − 9 m − 4 m + 6 = 6 m 2 − 13 m + 6 Thus, the expression is: 6 m 2 − 13 m + 6 m − 4 We should check if the numerator and denominator have any common factors to simplify further. In this case, they do not.
Final Answer Therefore, the expression 3 m − 2 2 − 2 m − 3 1 as a single fraction in its lowest terms is 6 m 2 − 13 m + 6 m − 4 .
Examples
Understanding how to combine fractions with algebraic expressions in the denominator is useful in various fields. For example, in electrical engineering, when analyzing circuits with impedances represented as complex fractions, you often need to combine these fractions to simplify the circuit analysis. Similarly, in physics, when dealing with wave interference or resonance phenomena, combining fractional expressions is essential to derive simplified equations that describe the system's behavior. This skill is also crucial in calculus when integrating rational functions using partial fraction decomposition.
To express 3 m − 2 2 − 2 m − 3 1 as a single fraction, find a common denominator, combine the fractions, simplify the numerator, and write the final expression. The result is 6 m 2 − 13 m + 6 m − 4 . This fraction is in its lowest terms.
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