Decide which of the ordered pairs are solutions for the equation y=4z+2. (4 points)

Question

Solution

Not A Solution

(-2-6)

(-1,0)

(-2,6)

(0, 2)

Answer 1

To determine which of the ordered pairs is a solution for the equation \( y = 4z + 2 \), we will substitute the \( z \) value from each pair into the equation and see if the resulting \( y \) value matches the \( y \) given in the pair.

1. For the pair \((-2, -6)\):

   • Substitute \( z = -2 \): \[ y = 4(-2) + 2 = -8 + 2 = -6 \]
   • This matches the \( y \) value of -6, so \((-2, -6)\) is a solution.

2. For the pair \((-1, 0)\):

   • Substitute \( z = -1 \): \[ y = 4(-1) + 2 = -4 + 2 = -2 \]
   • This does not match the \( y \) value of 0, so \((-1, 0)\) is not a solution.

3. For the pair \((-2, 6)\):

   • Substitute \( z = -2 \): \[ y = 4(-2) + 2 = -8 + 2 = -6 \]
   • This does not match the \( y \) value of 6, so \((-2, 6)\) is not a solution.

4. For the pair \((0, 2)\):

   • Substitute \( z = 0 \): \[ y = 4(0) + 2 = 0 + 2 = 2 \]
   • This matches the \( y \) value of 2, so \((0, 2)\) is a solution.

Summary:

• Solutions: \((-2, -6)\), \((0, 2)\)
• Not solutions: \((-1, 0)\), \((-2, 6)\)

Answer 2

Mrs. Wilbanks is taking a group of students to see a play. The cost of the bus rental to transport her class is $500 and the cost per ticket to see the play is $20. Create an equation with two variables that models this situation, then find the total cost for the event if 60 students & teachers plan to attend. (1 point)

OC500+201, $1,700

C500+ 20t. $1,700

OC500+20, $30,020

C-5201, $31,200

Answer 3

To model the situation with an equation involving two variables, we can define the variables as follows:

• Let \( C \) be the total cost for the event.
• Let \( t \) be the number of tickets (students and teachers) that need to be purchased.

The equation that models the total cost is: \[ C = 500 + 20t \] where:

• \( 500 \) is the fixed cost for the bus rental,
• \( 20t \) is the variable cost for the tickets, with each ticket costing $20.

Now, if 60 students and teachers plan to attend, we can substitute \( t = 60 \) into the equation:

\[ C = 500 + 20(60) \] Calculating this step-by-step:

1. Calculate \( 20 \times 60 = 1200 \).
2. Add the fixed cost: \( C = 500 + 1200 = 1700 \).

So, the total cost for the event if 60 students and teachers attend is $1,700.

The correct equation based on the options provided is: C = 500 + 20t, $1,700.

Answer 4

Sophia is selecting a planning committee for her school's annual talent show. The committee must consist of 12 students, with members from both the freshman and sophomore classes. At least two students must be from each class. Which of the following is a viable solution for this situation? (1 point)

1 freshman and 11 sophomores

6 freshmen and 4 sophomores

5 freshmen and 7 sophomores

O freshmen and 12 sophomores

Answer 5

To meet the requirements for Sophia's planning committee, we need to ensure the following:

1. The total number of committee members must be 12.
2. At least 2 students must be from the freshman class.
3. At least 2 students must be from the sophomore class.

Let's analyze each option based on these criteria:

1. 1 freshman and 11 sophomores:

   • Total: \(1 + 11 = 12\)
   • Freshmen: 1 (not meeting the requirement of at least 2)
   • Sophomores: 11
   • Not a viable solution.

2. 6 freshmen and 4 sophomores:

   • Total: \(6 + 4 = 10\) (this is incorrect, total must be 12)
   • Not a viable solution.

3. 5 freshmen and 7 sophomores:

   • Total: \(5 + 7 = 12\)
   • Freshmen: 5 (meets the requirement of at least 2)
   • Sophomores: 7 (meets the requirement of at least 2)
   • This is a viable solution.

4. 0 freshmen and 12 sophomores:

   • Total: \(0 + 12 = 12\)
   • Freshmen: 0 (not meeting the requirement of at least 2)
   • Sophomores: 12
   • Not a viable solution.

Conclusion:

The viable solution for the situation is 5 freshmen and 7 sophomores.

Answer 6

Alex won a gift card worth $96.00 for winning a design competition. She decided to spend her winnings on sketchbooks and paint sets. Each sketchbook (x) costs $12.00, while each paint set (y) costs $15.00.

Which of the following combinations of sketchbooks and paint sets can Alex purchase, assuming she spends her entire gift card?

(1 point)

3 sketchbooks and 4 paint sets

1 sketchbook and 4 paint sets

2 sketchbooks and 5 paint sets

©2

4 sketchbooks and 3 paint sets

Answer 7

To determine which combinations of sketchbooks and paint sets Alex can purchase with her $96 gift card, we will set up the equation based on the costs of each item:

Let:

• \( x \) = number of sketchbooks
• \( y \) = number of paint sets

The total cost of sketchbooks and paint sets can be represented by the equation: \[ 12x + 15y = 96 \]

Now, we will evaluate each of the provided combinations to see if they satisfy the equation.

1. 3 sketchbooks and 4 paint sets:

   • \( 12(3) + 15(4) = 36 + 60 = 96 \)
   • This combination works.

2. 1 sketchbook and 4 paint sets:

   • \( 12(1) + 15(4) = 12 + 60 = 72 \)
   • This combination does not work (total is $72).

3. 2 sketchbooks and 5 paint sets:

   • \( 12(2) + 15(5) = 24 + 75 = 99 \)
   • This combination does not work (total is $99).

4. 2 sketchbooks and 4 paint sets:

   • \( 12(2) + 15(4) = 24 + 60 = 84 \)
   • This combination does not work (total is $84).

5. 4 sketchbooks and 3 paint sets:

   • \( 12(4) + 15(3) = 48 + 45 = 93 \)
   • This combination does not work (total is $93).

Conclusion:

The only combination that Alex can purchase while spending her entire $96 gift card is 3 sketchbooks and 4 paint sets.