Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Question

Omar is going to send some flowers to his wife. Norwood Florist charges $2 per rose, plus $33 for the vase. Jane's Flowers, in contrast, charges $3 per rose and $11 for the vase. If Omar orders the bouquet with a certain number of roses, the cost will be the same with either flower shop. What would the total cost be? How many roses would there be?

The cost will be $
if the bouquet contains
roses.

Answer 1

Let's denote the number of roses Omar wants to buy as \( r \).

Cost from Norwood Florist: The cost for roses from Norwood Florist can be represented by the equation: \[ C_N = 2r + 33 \] This represents $2 per rose plus $33 for the vase.

Cost from Jane's Flowers: The cost for roses from Jane's Flowers can be represented by the equation: \[ C_J = 3r + 11 \] This represents $3 per rose plus $11 for the vase.

System of Equations: To find out how many roses will yield the same cost in both shops, we set the two equations equal to each other: \[ 2r + 33 = 3r + 11 \]

Solving the System using Substitution:

1. First, we can isolate \( r \) on one side of the equation.

   • Start by subtracting \( 2r \) from both sides: \[ 33 = r + 11 \]

2. Now, subtract 11 from both sides: \[ 33 - 11 = r \] \[ 22 = r \]

So, Omar would need to buy 22 roses.

Calculating Total Cost: Now we can substitute \( r = 22 \) back into either cost equation to find the total cost.

Using Norwood Florist's cost equation: \[ C_N = 2(22) + 33 \] \[ C_N = 44 + 33 \] \[ C_N = 77 \]

Using Jane's Flowers cost equation (for verification): \[ C_J = 3(22) + 11 \] \[ C_J = 66 + 11 \] \[ C_J = 77 \]

Both computations yield the same total cost, confirming our solution.

Thus, the final answers are: The cost will be $77 if the bouquet contains 22 roses.

So, filling in the blanks: The cost will be $77 if the bouquet contains 22 roses.