Which of the following expressions is equivalent to $(3 x)(4+y)-2.5 x$?
A. $12 x-0.5 x y$
B. $14.5 x+3 y$
C. $9.5 x+3 x y$
D. $7 x -2.5 y$
Answer (1)
Expand the expression: ( 3 x ) ( 4 + y ) − 2.5 x = 12 x + 3 x y − 2.5 x .
Combine like terms: 12 x − 2.5 x = 9.5 x .
The simplified expression is 9.5 x + 3 x y .
The equivalent expression is 9.5 x + 3 x y .
Explanation
Understanding the Problem We are given the expression ( 3 x ) ( 4 + y ) − 2.5 x and asked to find an equivalent expression from the options provided.
Expanding the Expression First, we need to expand the given expression by distributing 3 x to both terms inside the parentheses: ( 3 x ) ( 4 + y ) − 2.5 x = ( 3 x ) ( 4 ) + ( 3 x ) ( y ) − 2.5 x
Simplifying Terms Now, we simplify the terms: 12 x + 3 x y − 2.5 x
Combining Like Terms Next, we combine the like terms, which are 12 x and − 2.5 x :
12 x − 2.5 x = 9.5 x So the expression becomes: 9.5 x + 3 x y
Comparing with Options Finally, we compare our simplified expression 9.5 x + 3 x y with the given options: Option A: 12 x − 0.5 x y Option B: 14.5 x + 3 y Option C: 9.5 x + 3 x y Option D: 7 x − 2.5 y We see that Option C matches our simplified expression.
Final Answer Therefore, the expression equivalent to ( 3 x ) ( 4 + y ) − 2.5 x is 9.5 x + 3 x y .
Examples
Understanding equivalent expressions is crucial in various real-life scenarios, such as simplifying complex formulas in physics or engineering. For instance, when calculating the total cost of items with discounts and taxes, you often need to simplify expressions to make the calculations easier and more efficient. This skill is also valuable in financial planning, where you might need to manipulate algebraic expressions to understand investment returns or loan payments. By mastering the simplification of algebraic expressions, you can solve problems more effectively in many practical situations.
