Let's break down the problem and solve it step by step:

Problem Summary:

• The florist needs to make 5 identical bouquets, each containing 12 flowers.
• The total budget is $160.
• Flower prices:
  • Roses: $2.50 each.
  • Lilies: $4.00 each.
  • Irises: $2.00 each.
• The florist wants twice as many roses as the other two types of flowers combined.
• We need to determine how many of each type of flower will be in each bouquet, assuming the florist will use the entire budget.

Define Variables:

Let:

• $r$ = the number of roses in each bouquet.
• $l$ = the number of lilies in each bouquet.
• $i$ = the number of irises in each bouquet.

Step 1: Equation for the number of flowers in each bouquet

Each bouquet contains a total of 12 flowers: $r + l + i = 12$

Step 2: Condition for the number of roses

The number of roses is twice the sum of the number of lilies and irises combined: $r = 2(l + i)$

Step 3: Equation for the budget

The total budget is $160 for 5 bouquets, so the budget per bouquet is $\frac{160}{5} = 32$. Thus, the cost for each bouquet is: $2.50r + 4l + 2i = 32$

Step 4: Solve the system of equations

We now have three equations:

1. $r + l + i = 12$
2. $r = 2(l + i)$
3. $2.50r + 4l + 2i = 32$

I will now solve these equations.The solution to the system of equations is:

• 8 roses
• 2 lilies
• 2 irises

So, each bouquet should contain:

• 8 roses,
• 2 lilies,
• 2 irises.

This satisfies the conditions regarding the total number of flowers, the budget, and the ratio of roses to other flowers.

Would you like more details on any part of this solution, or have further questions? Here are some related questions to deepen your understanding:

1. How would the solution change if the budget was increased to $200?
2. What if the florist wanted three times as many roses as the other flowers combined?
3. How would the solution be affected if lilies and irises had the same price?
4. Can you generalize the system if the prices of flowers are different in each bouquet?
5. What would happen if the number of bouquets were increased to 10?

Tip:

When solving systems of equations, substitution or elimination methods are powerful tools to reduce the number of variables step by step.