Which situations can represent the expression 8−x ? Check the three that apply.(3 points) Responses Benjamin lost eight of his socks. Benjamin lost eight of his socks. Naomi had eight pencils and gave some away to her classmates. Naomi had eight pencils and gave some away to her classmates. Gabrielle decreased her eight-minute commute to school by some number of minutes. Gabrielle decreased her eight-minute commute to school by some number of minutes. Sydney increased her collection of stamps by eight. Sydney increased her collection of stamps by eight. Jones County has eight fewer elective courses than Smith County. Jones County has eight fewer elective courses than Smith County. Eight serving of lunch decreased by some number. Eight serving of lunch decreased by some number.

Question

At a farmers market, Samuel bought 1 pounds of apples for x dollars per pound and 3 bags of spinach for y dollars each. The next day, he returned and bought 6 pounds of apples for x dollars per pound and 2 bags of spinach for y dollars each. Which expression represents the total amount he spent at the market on both days?(1 point) Responses 4x+8y4x+8y4 x plus 8 y 7x+5y7x+5y7 x plus 5 y 7y+5x7y+5x7 y plus 5 x 4y+8x4y+8x4 y plus 8 x
Which pair shows equivalent expressions?(1 point) Responses −4x−12=−4(x+3)−4x−12=−4(x+3)negative 4 x minus 12 is equal to negative 4 times open paren x plus 3 close paren −4(x+3)=4x−12−4(x+3)=4x−12negative 4 times open paren x plus 3 close paren is equal to 4 x minus 12 4x+12=−4(x−3)4x+12=−4(x−3)4 x plus 12 is equal to negative 4 times open paren x minus 3 close paren −4(x−3)=−4x−12

Answer 1

Let's break down the questions one by one.

1. Situations that represent the expression \(8 - x\):

The expression \(8 - x\) indicates a situation where you start with 8 of something and then subtract a certain amount \(x\), representing something that has been lost, given away, or decreased.

• Benjamin lost eight of his socks. ➔ Not applicable, as this implies a negative effect but does not follow the structure of \(8 - x\); the amount is entirely lost.
• Naomi had eight pencils and gave some away to her classmates. ➔ Applicable. This represents \(8 - x\), where \(x\) is the number of pencils given away.
• Gabrielle decreased her eight-minute commute to school by some number of minutes. ➔ Applicable. This implies that her new time is \(8\) minus the number of minutes reduced.
• Sydney increased her collection of stamps by eight. ➔ Not applicable. This does not follow the \(8 - x\) structure.
• Jones County has eight fewer elective courses than Smith County. ➔ Applicable. This can be interpreted as Smith County having \(x\) elective courses, and Jones County having \(x - 8\).
• Eight servings of lunch decreased by some number. ➔ Applicable. This implies \(8 - x\), where \(x\) is the number of servings that were decreased.

Thus, the situations that apply are:

• Naomi had eight pencils and gave some away to her classmates.
• Gabrielle decreased her eight-minute commute to school by some number of minutes.
• Jones County has eight fewer elective courses than Smith County.
• Eight servings of lunch decreased by some number.

2. Expression representing the total amount Samuel spent:

Let's calculate the cost for Samuel on both days.

On the first day:

• He bought 1 pound of apples at \(x\) dollars per pound: Cost = \(1x\)
• He bought 3 bags of spinach at \(y\) dollars each: Cost = \(3y\)
• Total for the first day = \(1x + 3y = x + 3y\)

On the second day:

• He bought 6 pounds of apples at \(x\) dollars per pound: Cost = \(6x\)
• He bought 2 bags of spinach at \(y\) dollars each: Cost = \(2y\)
• Total for the second day = \(6x + 2y\)

Adding both days together: \[ \text{Total Cost} = (x + 3y) + (6x + 2y) = 7x + 5y \]

So, the expression that represents the total amount he spent at the market on both days is:

• 7x + 5y

3. Pair showing equivalent expressions:

Let's check the equations for equivalence.

1. \(-4x - 12 = -4(x + 3)\):

   • Simplifying the right side: \(-4(x + 3) = -4x - 12\)
   • Both sides are equal. ✅ (This pair is equivalent.)

2. \(-4(x + 3) = 4x - 12\):

   • Simplifying the left side: \(-4(x + 3) = -4x - 12\)
   • This does not equal \(4x - 12\). ❌ (Not equivalent.)

3. \(4x + 12 = -4(x - 3)\):

   • Simplifying the right side: \(-4(x - 3) = -4x + 12\)
   • This does not equal \(4x + 12\). ❌ (Not equivalent.)

4. \(-4(x - 3) = -4x - 12\):

   • Simplifying the left side: \(-4(x - 3) = -4x + 12\)
   • This does not equal \(-4x - 12\). ❌ (Not equivalent.)

Therefore, the equivalent expression is:

• \(-4x - 12 = -4(x + 3)\).

Summary:

1. Situations that represent \(8 - x\) include: Naomi's scenario, Gabrielle's scenario, Jones County, and the servings.
2. The expression representing Samuel's total amount spent: 7x + 5y.
3. The equivalent expressions are: \(-4x - 12 = -4(x + 3)\).