Question

part ii : a correct answer for question 18 will receive 4 credits. clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. utilize the information provided for each question to determine your answer. note that diagrams are not necessarily drawn to scale. for all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. all answers should be written in pen, except for graphs and drawings, which should be done in pencil.
18 alex had $1.70 in nickels and dimes on his desk. there were 25 coins in all. write a system of equations that could be used to determine both the number of nickels, n, and the number of dimes, d, that alex had.
use your system of equations to algebraically determine both the number of nickels and the number of dimes that he had.
part iii : a correct answer for question 19 will receive 6 credits. clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. utilize the information provided for each question to determine your answer, which should be done in pencil. all answers should be written in pen, except for graphs and drawings, which should be done in pencil.
19 anna plans to spend $30 on balloons and party hats for her daughter’s birthday party. including tax, balloons cost $2 each and party hats cost $1.50 each. the number of party hats anna needs is twice as many as the number of balloons. if x represents the number of balloons and y represents the number of party hats, write a system of equations that can be used to represent this situation.
graph your system of equations on the set of axes below.
state the coordinates of the point of intersection of your lines.
explain what each coordinate means in the context of the problem.

Explanation:

Question 18 (System of Equations for Coins)

Step 1: Define Variables and Set Up Equations

Let \( n \) = number of nickels, \( d \) = number of dimes.

• Total coins: \( n + d = 25 \) (since there are 25 coins total).
• Total value: Nickels are $0.05, dimes $0.10, total $1.70. So \( 0.05n + 0.10d = 1.70 \).

Step 2: Solve the System (Substitution or Elimination)

Use substitution: From \( n + d = 25 \), solve for \( n \): \( n = 25 - d \).

Substitute \( n = 25 - d \) into the value equation:

$$ 0.05(25 - d) + 0.10d = 1.70 $$

Step 3: Simplify and Solve for \( d \)

Expand: \( 1.25 - 0.05d + 0.10d = 1.70 \)
Combine like terms: \( 1.25 + 0.05d = 1.70 \)
Subtract 1.25: \( 0.05d = 0.45 \)
Divide by 0.05: \( d = \frac{0.45}{0.05} = 9 \)

Step 4: Solve for \( n \)

Substitute \( d = 9 \) into \( n = 25 - d \):
\( n = 25 - 9 = 16 \)

Question 19 (System of Equations for Party Supplies)

Step 1: Define Variables and Set Up Equations

Let \( x \) = number of balloons, \( y \) = number of party hats.

• Cost: Balloons cost $2, hats $1.50, total $30. So \( 2x + 1.50y = 30 \).
• Quantity: Hats are twice balloons: \( y = 2x \).

Step 2: Solve the System (Substitution)

Substitute \( y = 2x \) into the cost equation:

$$ 2x + 1.50(2x) = 30 $$

Step 3: Simplify and Solve for \( x \)

Expand: \( 2x + 3x = 30 \)
Combine like terms: \( 5x = 30 \)
Divide by 5: \( x = 6 \)

Step 4: Solve for \( y \)

Substitute \( x = 6 \) into \( y = 2x \):
\( y = 2(6) = 12 \)

Step 5: Graph and Interpret Intersection

• Equation 1 (cost): \( 2x + 1.50y = 30 \) → Solve for \( y \): \( y = \frac{30 - 2x}{1.50} = 20 - \frac{4}{3}x \).
• Equation 2 (quantity): \( y = 2x \).

Graph both lines. Their intersection is at \( (x, y) = (6, 12) \).

• \( x = 6 \): Number of balloons (6 balloons).
• \( y = 12 \): Number of party hats (12 hats, which is twice 6 balloons).

Final Answers:

Question 18:

• System of equations: \( \boldsymbol{n + d = 25} \) and \( \boldsymbol{0.05n + 0.10d = 1.70} \).
• Number of nickels: \( \boldsymbol{16} \), number of dimes: \( \boldsymbol{9} \).

Question 19:

• System of equations: \( \boldsymbol{2x + 1.50y = 30} \) and \( \boldsymbol{y = 2x} \).
• Intersection: \( \boldsymbol{(6, 12)} \) (6 balloons, 12 party hats).

Answer:

Question 18 (System of Equations for Coins)

Step 1: Define Variables and Set Up Equations

Let \( n \) = number of nickels, \( d \) = number of dimes.

• Total coins: \( n + d = 25 \) (since there are 25 coins total).
• Total value: Nickels are $0.05, dimes $0.10, total $1.70. So \( 0.05n + 0.10d = 1.70 \).

Step 2: Solve the System (Substitution or Elimination)

Use substitution: From \( n + d = 25 \), solve for \( n \): \( n = 25 - d \).

Substitute \( n = 25 - d \) into the value equation:

$$ 0.05(25 - d) + 0.10d = 1.70 $$

Step 3: Simplify and Solve for \( d \)

Expand: \( 1.25 - 0.05d + 0.10d = 1.70 \)
Combine like terms: \( 1.25 + 0.05d = 1.70 \)
Subtract 1.25: \( 0.05d = 0.45 \)
Divide by 0.05: \( d = \frac{0.45}{0.05} = 9 \)

Step 4: Solve for \( n \)

Substitute \( d = 9 \) into \( n = 25 - d \):
\( n = 25 - 9 = 16 \)

Question 19 (System of Equations for Party Supplies)

Step 1: Define Variables and Set Up Equations

Let \( x \) = number of balloons, \( y \) = number of party hats.

• Cost: Balloons cost $2, hats $1.50, total $30. So \( 2x + 1.50y = 30 \).
• Quantity: Hats are twice balloons: \( y = 2x \).

Step 2: Solve the System (Substitution)

Substitute \( y = 2x \) into the cost equation:

$$ 2x + 1.50(2x) = 30 $$

Step 3: Simplify and Solve for \( x \)

Expand: \( 2x + 3x = 30 \)
Combine like terms: \( 5x = 30 \)
Divide by 5: \( x = 6 \)

Step 4: Solve for \( y \)

Substitute \( x = 6 \) into \( y = 2x \):
\( y = 2(6) = 12 \)

Step 5: Graph and Interpret Intersection

• Equation 1 (cost): \( 2x + 1.50y = 30 \) → Solve for \( y \): \( y = \frac{30 - 2x}{1.50} = 20 - \frac{4}{3}x \).
• Equation 2 (quantity): \( y = 2x \).

Graph both lines. Their intersection is at \( (x, y) = (6, 12) \).

• \( x = 6 \): Number of balloons (6 balloons).
• \( y = 12 \): Number of party hats (12 hats, which is twice 6 balloons).

Final Answers:

Question 18:

• System of equations: \( \boldsymbol{n + d = 25} \) and \( \boldsymbol{0.05n + 0.10d = 1.70} \).
• Number of nickels: \( \boldsymbol{16} \), number of dimes: \( \boldsymbol{9} \).

Question 19:

• System of equations: \( \boldsymbol{2x + 1.50y = 30} \) and \( \boldsymbol{y = 2x} \).
• Intersection: \( \boldsymbol{(6, 12)} \) (6 balloons, 12 party hats).