To find out how many carnations, roses, and daisies were ordered, let's define some variables and set up a system of equations:

Step 1: Define the variables

Let:

• $c$ be the number of carnations ordered.
• $r$ be the number of roses ordered.
• $d$ be the number of daisies ordered.

Step 2: Write equations based on the problem

1. Total number of flowers:
   Billy's restaurant ordered 200 flowers, so: $c + r + d = 200$

2. Price equation:
   The total cost of the flowers is $589.50. The carnations cost $1.50 each, roses cost $5.75 each, and daisies cost $2.60 each. Therefore, we can write the total price equation as: $1.50c + 5.75r + 2.60d = 589.50$

3. Relationship between roses and daisies:
   The problem says they ordered 20 fewer roses than daisies, so: $r = d - 20$

Step 3: Substitute and solve the system of equations

We now have three equations:

1. $c + r + d = 200$
2. $1.50c + 5.75r + 2.60d = 589.50$
3. $r = d - 20$

Start by substituting $r = d - 20$ into the first two equations.

Equation 1:

$c + (d - 20) + d = 200$ Simplifying: $c + 2d - 20 = 200 \quad \text{or} \quad c + 2d = 220$ So we get: $c = 220 - 2d \quad \text{(Equation 4)}$

Equation 2:

Substitute $r = d - 20$ into the price equation: $1.50c + 5.75(d - 20) + 2.60d = 589.50$ Simplify this: $1.50c + 5.75d - 115 + 2.60d = 589.50$ $1.50c + 8.35d = 704.50 \quad \text{(Equation 5)}$

Step 4: Substitute $c = 220 - 2d$ into Equation 5

Substitute Equation 4 into Equation 5: $1.50(220 - 2d) + 8.35d = 704.50$ Simplifying: $330 - 3d + 8.35d = 704.50$ $330 + 5.35d = 704.50$ $5.35d = 374.50$ $d = \frac{374.50}{5.35} \approx 70$

Step 5: Find $r$ and $c$

Now that we know $d \approx 70$, we can find $r$ and $c$:

• $r = d - 20 = 70 - 20 = 50$
• $c = 220 - 2d = 220 - 2(70) = 80$

Final Answer:

• Carnations: $80$
• Roses: $50$
• Daisies: $70$

These are the number of each type of flower ordered.

Would you like a breakdown of any specific step?

Here are 5 related questions for practice:

1. What would be the total cost if the price of daisies increased by $1 each?
2. If the total number of flowers ordered was 250, how would the solution change?
3. What if the restaurant ordered 10 more roses than daisies instead of 20 fewer?
4. How would the system change if the cost of carnations was $2 each?
5. What if the total cost of the flowers was $700 instead of $589.50?

Tip: Solving systems of equations step-by-step, especially with substitution, helps to avoid mistakes in calculations.