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At a farmers market, Samuel bought 3 pounds of apples for x dollars per pound and 2 bags of spinach for y dollars each. The next day he returned and bought 5 pounds of apples for x dollars per pound and 3 bags of spinach for y dollars each. Which expression represents the total amount he spent at the market on both days?(1 point) Responses 6x+7y6x+7y6 x plus 7 y 8y+5x8y+5x8 y plus 5 x 6y+7x6y+7x6 y plus 7 x 8x+5y8x+5y8 x plus 5 y

Which expression is equivalent to 30(12x−2)+40(34y−4)30(12x−2)+40(34y−4) (1 point) Responses 15x+30y−22015x+30y−22015 x plus 30 y minus 220 15x+30y−6415x+30y−6415 x plus 30 y minus 64 15x−30y−22015x−30y−22015 x minus 30 y minus 220 45xy−220
Which pair shows equivalent expressions?(1 point) Responses 2x+10=−2(x−5)2x+10=−2(x−5)2 x plus 10 is equal to negative 2 times open paren x minus 5 close paren −2(x+5)=2x−10−2(x+5)=2x−10negative 2 times open paren x plus 5 close paren is equal to 2 x minus 10 −2(x−5)=−2x−10−2(x−5)=−2x−10negative 2 times open paren x minus 5 close paren is equal to negative 2 x minus 10 −2x−10=−2(x+5)
This morning, Sam walked 4 kilometers in 50 minutes. At what rate did Sam walk? Use the formula r=dtr=dt where r, is the rate, d is the distance, and t is the time. (1 point) Responses 12.5 kilometers per minute 12.5 kilometers per minute 0.08 kilometers per minute 0.08 kilometers per minute 46 kilometers per minute 46 kilometers per minute 200 kilometers per minute 200 kilometers per minute

1 answer

Let's break down each question step by step.

  1. Total Amount Spent at the Market:

    On the first day, Samuel bought:

    • 3 pounds of apples at \( x \) dollars per pound: \( 3x \)
    • 2 bags of spinach at \( y \) dollars each: \( 2y \)

    Total spent on the first day: \( 3x + 2y \)

    On the second day, he bought:

    • 5 pounds of apples at \( x \) dollars per pound: \( 5x \)
    • 3 bags of spinach at \( y \) dollars each: \( 3y \)

    Total spent on the second day: \( 5x + 3y \)

    Therefore, the total amount spent over both days is: \[ (3x + 2y) + (5x + 3y) = 3x + 5x + 2y + 3y = 8x + 5y \]

    The correct expression is: \( 8x + 5y \)

  2. Equivalent Expression for \( 30(12x−2)+40(34y−4) \):

    Distributing each term gives:

    For the first part: \[ 30(12x - 2) = 360x - 60 \]

    For the second part: \[ 40(34y - 4) = 1360y - 160 \]

    Combining these results: \[ (360x - 60) + (1360y - 160) = 360x + 1360y - 220 \]

    So, the equivalent expression is: \( 360x + 1360y - 220 \), which is not listed as a choice. Let's confirm by simplifying each option against the original expression.

  3. Equivalent Expressions:

    Evaluate each pair of expressions:

    • First pair: \( 2x + 10 = -2(x - 5) \)

      Distributing on the right: \(-2(x - 5) = -2x + 10\), giving: \( 2x + 10 = -2x + 10\) does not hold true.

    • Second pair: \(-2(x + 5) = 2x - 10\)

      Distributing on the left: \(-2(x + 5) = -2x - 10\) gives \( -2x - 10 = 2x - 10\), does not hold true.

    • Third pair: \(-2(x - 5) = -2x - 10\)

      Distributing on the left: \(-2(x - 5) = -2x + 10\) gives \( -2x + 10 = -2x - 10\), does **not hold true.

    • Fourth pair: \(-2x - 10 = -2(x + 5)\)

      Distributing on the right: \(-2(x + 5) = -2x - 10\) gives \( -2x - 10 = -2x - 10\), holds true.

    The equivalent expressions are: \( -2x - 10 = -2(x + 5) \)

  4. Sam's Walking Rate:

    Using the formula \( r = \frac{d}{t} \):

    • Distance \( d = 4 \) kilometers
    • Time \( t = 50 \) minutes

    Calculate the rate: \[ r = \frac{4 \text{ km}}{50 \text{ min}} = 0.08 \text{ km/min} \]

    Regularizing into a possible metric gives:

    • \( 0.08 \) kilometers per minute

    The correct rate is: \( 0.08 \) kilometers per minute.

To summarize the answers:

  1. \( 8x + 5y \)
  2. None of the options provided matches exactly after simplification.
  3. \( -2x - 10 = -2(x + 5) \)
  4. \( 0.08 \) kilometers per minute