The florist charge $31.75 for eight roses and five carnations. For one rose and three carnations,it costs $5.75. What is the cost for each type of flower?

Let's use the variable x to represent the cost of one rose and y to represent the cost of one carnation.

If 8 roses and 5 carnations cost $31.75, we can write the following equation:

$8x+5y=31.75$

If 1 rose and 3 carnations cost $5.75, we can write a second equation:

$x+3y=5.75$

From the second equation, we have x = 5.75 - 3y.

Using this value of x in the first equation, we have:

$\begin{gathered} 8(5.75-3y)+5y=31.75\\ \\ 46-24y+5y=31.75\\ \\ -19y=31.75-46\\ \\ -19y=-14.25\\ \\ y=0.75 \end{gathered}$

Now, calculating the cost of one rose, we have:

$x=5.75-3y=5.75-3\cdot0.75=5.75-2.25=3.5$

Therefore the cost of one rose is $3.50 and the cost of one carnation is $0.75.

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