Select the correct answer.
Which is the simplified form of the expression $3\left(\frac{7}{5} x+4\right)-2\left(\frac{3}{2}-\frac{5}{4} x\right) ?$
A. $-\frac{39}{5} x-\frac{11}{2}$
B. $\frac{67}{10} x+9$
C. $\frac{3}{10} x+\frac{5}{2}$
D. $15+\frac{76}{10} x$
Answer (2)
To simplify the given expression:
Distribute the constants: 3 ( 5 7 x + 4 ) = 5 21 x + 12 and − 2 ( 2 3 − 4 5 x ) = − 3 + 2 5 x .
Combine the expressions: ( 5 21 x + 12 ) + ( − 3 + 2 5 x ) .
Combine like terms: ( 5 21 x + 2 5 x ) + ( 12 − 3 ) .
Simplify: 10 67 x + 9 .
The correct answer is 10 67 x + 9 .
Explanation
Understanding the problem We are asked to simplify the expression 3\[\frac{7}{5} x+4\]-2\[\frac{3}{2}-\frac{5}{4} x\] and choose the correct answer from the given options.
Distributing the constants First, distribute the constants into the parentheses:
3 ( 5 7 x + 4 ) = 3 × 5 7 x + 3 × 4 = 5 21 x + 12
− 2 ( 2 3 − 4 5 x ) = − 2 × 2 3 − 2 × ( − 4 5 x ) = − 3 + 4 10 x = − 3 + 2 5 x
Combining the expressions Now, combine the two resulting expressions:
( 5 21 x + 12 ) + ( − 3 + 2 5 x )
Combining like terms Combine like terms (x terms and constant terms):
( 5 21 x + 2 5 x ) + ( 12 − 3 )
Adding x terms Find a common denominator for the x terms:
5 21 x + 2 5 x = 5 × 2 21 × 2 x + 2 × 5 5 × 5 x = 10 42 x + 10 25 x = 10 42 + 25 x = 10 67 x
Adding constants Combine the constant terms:
12 − 3 = 9
Simplified expression Write the simplified expression:
10 67 x + 9
Final Answer Compare the simplified expression with the given options and select the correct answer. The simplified form of the expression is 10 67 x + 9 , which corresponds to option B.
Examples
Simplifying expressions is a fundamental skill in algebra and is used in many real-world applications. For example, suppose you are calculating the total cost of materials for a project. You might have an expression that represents the cost of wood plus the cost of nails, and you need to simplify it to find the total cost. Simplifying algebraic expressions helps in optimizing resources and making informed decisions in various fields like engineering, economics, and computer science. It allows us to model real-world situations mathematically and find efficient solutions.
The expression 3 ( 5 7 x + 4 ) − 2 ( 2 3 − 4 5 x ) simplifies to 10 67 x + 9 . Therefore, the correct answer is option B. This process involves distributing the constants, combining like terms, and ensuring that all parts are correctly simplified.
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