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)
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 to get the final answer: 10 67 x + 9 .
Explanation
Understanding the Expression We are given the expression 3\tleft(\frac{7}{5} x+4\right)-2\left(\frac{3}{2}-\frac{5}{4} x\right) and we want to simplify it.
Distributing 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 Expressions Now, combine the two resulting expressions:
( 5 21 x + 12 ) + ( − 3 + 2 5 x )
Combining Like Terms Next, combine like terms (x terms and constant terms):
( 5 21 x + 2 5 x ) + ( 12 − 3 )
Finding Common Denominator Find a common denominator for the x terms:
5 21 x + 2 5 x = 10 42 x + 10 25 x = 10 67 x
Combining Constant Terms 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. The correct answer is:
10 67 x + 9
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with numerous real-world applications. For instance, consider a scenario where you are planning a rectangular garden. You want to determine the total amount of fencing needed based on variable side lengths. If the length of the garden is represented by 5 7 x + 4 and you need three sections of this length, and the width is represented by 2 3 − 4 5 x but you only need two sections of this width, the simplified expression 10 67 x + 9 helps you quickly calculate the total fencing required without having to individually compute each section's length every time you change the value of x .
The simplified form of the expression 3 ( 5 7 x + 4 ) − 2 ( 2 3 − 4 5 x ) is 10 67 x + 9 , which corresponds to option B. We achieved this by distributing the constants, combining like terms, and simplifying the result. Therefore, the answer is B.
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