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Which situations can represent the expression 6−x ? Check the three that apply.(3 points) Responses Westville has 6 fewer schools than Eastville. Westville has 6 fewer schools than Eastville. - only 3 answers are allowed Six servings of dinner were decreased by a number. Six servings of dinner were decreased by a number. - only 3 answers are allowed Naomi gives some of her six pencils away. Naomi gives some of her six pencils away. - only 3 answers are allowed Sydney increased her collection of coins by six. Sydney increased her collection of coins by six. - only 3 answers are allowed Gabrielle decreased her 6-minute mile by an unknown amount of time. Gabrielle decreased her 6-minute mile by an unknown amount of time. - only 3 answers are allowed Benjamin lost six of his stickers.

5 answers

To identify the situations that can represent the expression \( 6 - x \), we should look for examples where we start with 6 of something and then subtract some unknown quantity \( x \).

Here are the relevant situations:

  1. Six servings of dinner were decreased by a number: This can be represented as \( 6 - x \), where \( x \) is the number by which the servings were decreased.

  2. Naomi gives some of her six pencils away: This also represents \( 6 - x \), where \( x \) is the number of pencils she gave away.

  3. Gabrielle decreased her 6-minute mile by an unknown amount of time: This can be expressed as \( 6 - x \), with \( x \) being the time decreased from her mile.

The other statements do not represent the expression \( 6 - x \):

  • "Westville has 6 fewer schools than Eastville" doesn't translate into \( 6 - x \) because it doesn't start with 6.
  • "Sydney increased her collection of coins by six" indicates a situation where we add to 6, rather than subtract.
  • "Benjamin lost six of his stickers" indicates a loss of a fixed amount (6), rather than reducing from a starting point of 6.

Thus, the three appropriate responses are:

  1. Six servings of dinner were decreased by a number.
  2. Naomi gives some of her six pencils away.
  3. Gabrielle decreased her 6-minute mile by an unknown amount of time.
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 8x+5y 8 x plus 5 y 6x+7y 6 x plus 7 y 8y+5x 8 y plus 5 x 6y+7x 6 y plus 7 x

To find the total amount Samuel spent at the market on both days, we need to break down the costs for each day.

First day:

  • Samuel bought 3 pounds of apples for \( x \) dollars per pound: \[ 3x , \text{(cost of apples)} \]
  • He bought 2 bags of spinach for \( y \) dollars each: \[ 2y , \text{(cost of spinach)} \]

So, the total cost for the first day is: \[ 3x + 2y \]

Second day:

  • Samuel bought 5 pounds of apples for \( x \) dollars per pound: \[ 5x , \text{(cost of apples)} \]
  • He bought 3 bags of spinach for \( y \) dollars each: \[ 3y , \text{(cost of spinach)} \]

So, the total cost for the second day is: \[ 5x + 3y \]

Total amount spent over both days: Now we combine the totals from both days: \[ (3x + 2y) + (5x + 3y) = 3x + 5x + 2y + 3y = 8x + 5y \]

Therefore, the expression that represents the total amount Samuel spent at the market on both days is: \[ \boxed{8x + 5y} \]

After simplifying, which expressions are equivalent? Check the three that apply.(3 points) Responses 5.9a−5.6b 5 point 9 A minus 5 point 6 b (−3.9b+a)+(−1.7b+4.9a) open paren negative 3 point 9 b plus A close paren plus open paren negative 1 point 7 b plus 4 point 9 A close paren −0.4b+(6b−5.9a) negative 0 point 4 b plus open paren 6 b minus 5 point 9 A close paren (3.4a−1.7b)+(2.5a−3.9b) open paren 3 point 4 A minus 1 point 7 b close paren plus open paren 2 point 5 A minus 3 point 9 b close paren (2.5a+1.6b)+(3.4a+4b)

To determine which expressions are equivalent after simplifying, we'll simplify each of the expressions provided and see which ones yield the same result.

  1. Expression: \( 5.9a - 5.6b \)

This expression is already simplified.

  1. Expression: \( (-3.9b + a) + (-1.7b + 4.9a) \)

Simplifying this: \[ (-3.9b - 1.7b) + (a + 4.9a) = (-3.9 - 1.7)b + (1 + 4.9)a = -5.6b + 5.9a \]

  1. Expression: \( -0.4b + (6b - 5.9a) \)

Simplifying this: \[ -0.4b + 6b - 5.9a = (6 - 0.4)b - 5.9a = 5.6b - 5.9a \] Rearranging gives: \( -5.9a + 5.6b \), which is equivalent to \( 5.9a - 5.6b \) when rearranged.

  1. Expression: \( (3.4a - 1.7b) + (2.5a - 3.9b) \)

Simplifying this: \[ (3.4a + 2.5a) + (-1.7b - 3.9b) = (3.4 + 2.5)a + (-1.7 - 3.9)b = 5.9a - 5.6b \]

  1. Expression: \( (2.5a + 1.6b) + (3.4a + 4b) \)

Simplifying this: \[ (2.5a + 3.4a) + (1.6b + 4b) = (2.5 + 3.4)a + (1.6 + 4)b = 5.9a + 5.6b \]

Now, we can summarize the simplified versions of the expressions:

  • 1: \( 5.9a - 5.6b \)
  • 2: \( -5.6b + 5.9a \) (or \( 5.9a - 5.6b \))
  • 3: \( 5.6b - 5.9a \) (not equivalent)
  • 4: \( 5.9a - 5.6b \)
  • 5: \( 5.9a + 5.6b \) (not equivalent)

Summary of Equivalent Expressions

The three equivalent expressions are:

  1. \( 5.9a - 5.6b \)
  2. \( (-3.9b + a) + (-1.7b + 4.9a) \)
  3. \( (3.4a - 1.7b) + (2.5a - 3.9b) \)

Thus, the three that apply are:

  • \( 5.9a - 5.6b \)
  • \( (-3.9b + a) + (-1.7b + 4.9a) \)
  • \( (3.4a - 1.7b) + (2.5a - 3.9b) \)