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Equivalent expressions SAT question

Math · Advanced Math · Medium · Answer it here, explanation included.

If $p equals, 3 x, plus 4$ and $v equals, x plus 5$ , which of the following is equivalent to $p v, minus 2 p, plus v$ ?

ExplanationShow

Choice B is correct. It’s given that $p equals, 3 x plus 4$ and $v equals, x plus 5$ . Substituting the values for p and v into the expression $p v, minus 2 p, plus v$ yields $open parenthesis, 3 x plus 4, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, minus, 2 times, open parenthesis, 3 x plus 4, close parenthesis, plus x, plus 5$ . Multiplying the terms $open parenthesis, 3 x plus 4, close parenthesis, times, open parenthesis, x plus 5, close parenthesis$ yields $3 x squared, plus 4 x, plus 15 x, plus 20$ . Using the distributive property to rewrite $negative 2 times, open parenthesis, 3 x plus 4, close parenthesis$ yields $negative 6 x minus 8$ . Therefore, the entire expression can be represented as $3 x squared, plus 4 x, plus 15 x, plus 20, minus 6 x, minus 8, plus x, plus 5$ . Combining like terms yields $3 x squared, plus 14 x, plus 17$ .

Choice A is incorrect and may result from subtracting, instead of adding, the term $x plus 5$ . Choice C is incorrect. This is the result of multiplying the terms $open parenthesis, 3 x plus 4, close parenthesis, times, open parenthesis, x plus 5, close parenthesis$ . Choice D is incorrect and may result from distributing 2, instead of $negative 2$ , to the term $3 x plus 4$ .

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