Let’s analyze each given expression to determine if it is equivalent to "the quotient of 10 plus [tex]\( x \)[/tex] and [tex]\( y \)[/tex] minus 3".
1. [tex]\(\frac{10 + x}{y} - 3\)[/tex]
This expression follows the order of operations (PEMDAS/BODMAS). The quotient of [tex]\(10 + x\)[/tex] and [tex]\( y \)[/tex] is calculated first, and then 3 is subtracted. This matches exactly with "the quotient of 10 plus [tex]\( x \)[/tex] and [tex]\( y \)[/tex] minus 3".
Therefore, this expression is equivalent.
2. [tex]\(\frac{10 + x}{y - 3}\)[/tex]
In this expression, the subtraction of 3 is from the denominator [tex]\( y \)[/tex]. So, the expression is calculating the quotient of [tex]\(10 + x\)[/tex] and [tex]\( y - 3 \)[/tex], which is different from the provided phrase.
Therefore, this expression is not equivalent.
3. [tex]\(10 + \frac{x}{y} - 3\)[/tex]
Here, first [tex]\( x \)[/tex] is divided by [tex]\( y \)[/tex], then 10 is added to the result, and finally, 3 is subtracted from it. This sequence of operations does not match "the quotient of 10 plus [tex]\( x \)[/tex] and [tex]\( y \)[/tex] minus 3".
Therefore, this expression is not equivalent.
4. [tex]\(\frac{10 + x - 3}{y}\)[/tex]
This expression simplifies to [tex]\(\frac{7 + x}{y}\)[/tex], which means it calculates the quotient of [tex]\( 7 + x \)[/tex] and [tex]\( y \)[/tex], differing from the required "the quotient of 10 plus [tex]\( x \)[/tex] and [tex]\( y \)[/tex] minus 3".
Therefore, this expression is not equivalent.
Summarizing, only the first expression [tex]\(\frac{10 + x}{y} - 3\)[/tex] is equivalent to "the quotient of 10 plus [tex]\( x \)[/tex] and [tex]\( y \)[/tex] minus 3". The detailed analysis yields the following results:
1. [tex]\(\frac{10 + x}{y} - 3\)[/tex] is equivalent. (True)
2. [tex]\(\frac{10 + x}{y - 3}\)[/tex] is not equivalent. (False)
3. [tex]\(10 + \frac{x}{y} - 3\)[/tex] is not equivalent. (False)
4. [tex]\(\frac{10 + x - 3}{y}\)[/tex] is not equivalent. (False)
