Problem Setup:

Let $r$ represent the number of roses. The cost from Summerfield Florist and Polly's Flowers can be represented as follows:

1. Summerfield Florist: $C_1 = 3r + 21$
2. Polly's Flowers: $C_2 = 2r + 31$

We are told that the costs are the same for a certain number of roses, so: $3r + 21 = 2r + 31$

Solve Using Substitution:

1. Simplify the equation: $3r - 2r = 31 - 21$ $r = 10$

2. Substitute $r = 10$ into one of the cost equations (e.g., $C_1$): $C_1 = 3(10) + 21 = 30 + 21 = 51$

Thus, $r = 10$ and the total cost is $C = 51$.

Final Answer:

• Number of roses: $\mathbf{10}$
• Total cost: $\mathbf{51}$

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Would you like further clarification on this solution?

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Related Questions:

1. What if Summerfield Florist reduced their vase price to $15? How would the system of equations change?
2. What is the difference in cost for 5 roses between the two florists?
3. How would you set up the system if Polly’s Flowers charged $2.50 per rose instead?
4. What is the breakeven point in cost if Summerfield Florist increased the vase price by $5?
5. If Zachary wants to spend less than $50, how many roses can he afford from each florist?

Tip:

When solving systems of equations, always verify your solution by substituting back into the original equations to ensure consistency.