Simplify $\frac{6 x y z}{2 x y-y} \div \frac{2 x^2-7 x+3}{3 x z-9 z}$.
A. $\frac{18 x z^2}{4 x^2-4 x+1}$
B. $\frac{9 z^2}{4 x^2-1}$
C. $\frac{4 y}{3 x^2-7}$
D. $\frac{1}{2 x}$
Answer (2)
The expression simplifies to 4 x 2 − 4 x + 1 18 x z 2 , which matches option A. The simplification involved rewriting division as multiplication by the reciprocal, factoring, canceling common terms, and multiplying the remaining parts. Thus, the correct answer is option A.
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Rewrite the division as multiplication by the reciprocal: 2 x y − y 6 x yz ⋅ 2 x 2 − 7 x + 3 3 x z − 9 z .
Factor out common terms: y ( 2 x − 1 ) 6 x yz ⋅ ( 2 x − 1 ) ( x − 3 ) 3 z ( x − 3 ) .
Cancel out common factors: 2 x − 1 6 x z ⋅ 2 x − 1 3 z .
Multiply the remaining terms and expand: 4 x 2 − 4 x + 1 18 x z 2 .
The simplified expression matches option A: 4 x 2 − 4 x + 1 18 x z 2 .
Explanation
Problem Analysis We are asked to simplify the expression 2 x y − y 6 x yz ÷ 3 x z − 9 z 2 x 2 − 7 x + 3 . This involves dividing two rational expressions, which can be simplified by factoring and canceling common factors.
Rewrite as Multiplication First, we rewrite the division as multiplication by the reciprocal: 2 x y − y 6 x yz ÷ 3 x z − 9 z 2 x 2 − 7 x + 3 = 2 x y − y 6 x yz ⋅ 2 x 2 − 7 x + 3 3 x z − 9 z
Factorization Next, we factor out common terms in the numerators and denominators: y ( 2 x − 1 ) 6 x yz ⋅ ( 2 x − 1 ) ( x − 3 ) 3 z ( x − 3 )
Cancellation Now, we cancel out common factors: y ( 2 x − 1 ) 6 x yz ⋅ ( 2 x − 1 ) ( x − 3 ) 3 z ( x − 3 ) = 2 x − 1 6 x z ⋅ 2 x − 1 3 z
Multiplication Multiply the remaining terms: 2 x − 1 6 x z ⋅ 2 x − 1 3 z = ( 2 x − 1 ) 2 18 x z 2
Expansion Expand the denominator: ( 2 x − 1 ) 2 18 x z 2 = 4 x 2 − 4 x + 1 18 x z 2
Final Answer Finally, we compare the simplified expression with the given options. The simplified expression is 4 x 2 − 4 x + 1 18 x z 2 , which matches option A.
Examples
Rational expressions are used in various fields, such as physics, engineering, and economics, to model relationships between different quantities. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves. Simplifying rational expressions makes it easier to analyze and understand these relationships.
