Which expression is equivalent to $3(x-6)+5(x-4)$?
A) $9 x-22$
B) $8 x-38$
C) $8 x-10$
D) $15 x^2-38 x$
Answer (2)
Distribute the constants into the parentheses: 3 ( x − 6 ) = 3 x − 18 and 5 ( x − 4 ) = 5 x − 20 .
Combine the two resulting expressions: ( 3 x − 18 ) + ( 5 x − 20 ) .
Combine like terms: 3 x + 5 x = 8 x and − 18 − 20 = − 38 .
The simplified expression is 8 x − 38 .
Explanation
Analyze the problem We are given the expression 3 ( x − 6 ) + 5 ( x − 4 ) and asked to find an equivalent expression from the following options: A) 9 x − 22 B) 8 x − 38 C) 8 x − 10 D) 15 x 2 − 38 x
Plan the solution To find the equivalent expression, we need to distribute the constants into the parentheses and then combine like terms.
Distribute the first term First, distribute the 3 into ( x − 6 ) : 3 ( x − 6 ) = 3 x − 3 ( 6 ) = 3 x − 18
Distribute the second term Next, distribute the 5 into ( x − 4 ) : 5 ( x − 4 ) = 5 x − 5 ( 4 ) = 5 x − 20
Combine the expressions Now, combine the two resulting expressions: ( 3 x − 18 ) + ( 5 x − 20 )
Combine like terms Combine like terms. The x terms are 3 x and 5 x , which combine to 3 x + 5 x = 8 x . The constant terms are − 18 and − 20 , which combine to − 18 − 20 = − 38 .
Simplified expression Therefore, the simplified expression is 8 x − 38 .
Find the matching option Comparing the simplified expression 8 x − 38 with the given options, we see that it matches option B.
Final Answer Thus, the equivalent expression is 8 x − 38 .
Examples
In algebra, simplifying expressions is a fundamental skill. For example, if you're calculating the total cost of buying 'x' number of items, where each item costs $3 and has a discount of $6, and then adding that to the cost of buying 'x' number of another item costing $5 each with a discount of $4, the expression 3(x-6) + 5(x-4) represents the total cost. Simplifying this expression to 8x - 38 allows for easier calculation of the total cost for any number of items 'x'.
The equivalent expression to 3 ( x − 6 ) + 5 ( x − 4 ) simplifies to 8 x − 38 , which matches option B. This was achieved by distributing the constants and combining like terms. Thus, the final answer is 8 x − 38 .
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