Sandra is buying cupcakes and cake slices for her party. At the bakery she goes to, each cupcake costs $2.50 and each cake slice costs $3.25. Sandra buys a total of 22 items and spends $62.50. Write and solve a system of equations to determine how many cupcakes and how many cake slices Sandra bought. Be sure to define your variables.
Let:
x = number of cupcakes purchased
y = number of cake slices purchased
The total number of items purchased is 22:
x + y = 22
The total amount spent is $62.50:
2.50x + 3.25y = 62.50
Now we have the system of equations:
x + y = 22
2.50x + 3.25y = 62.50
We can solve this system of equations by substituting the value of x from the first equation into the second equation:
x = 22 - y
2.50(22 - y) + 3.25y = 62.50
55 - 2.50y + 3.25y = 62.50
0.75y = 7.50
y = 10
Now that we have found y, we can substitute it back into the first equation to find x:
x + 10 = 22
x = 12
Therefore, Sandra bought 12 cupcakes and 10 cake slices.