Let's start by assigning variables to the unknown quantities.
Let:
Y = number of cakes sold
X = total sales and expenditures
The cost of the booth is $52, and the cost of ingredients for each cake is $3. Therefore, the cost of ingredients for Y cakes is 3Y.
The revenue from selling each cake is $16. Therefore, the total revenue from selling Y cakes is 16Y.
The equation to represent the sales and expenditures is:
X = total revenue - total cost = (16Y) - (3Y + 52)
Now we can solve for Y by substituting the value of X:
X = (16Y) - (3Y + 52)
X = 16Y - 3Y - 52
X = 13Y - 52
Since we want to find both X and Y, we need another equation.
The total sales and expenditures X is equal to the revenue from selling Y cakes plus the cost of the booth:
X = 16Y + 52
Now we can solve this system of equations using the substitution method:
Step 1: Rewrite one equation in terms of one variable.
Using the equation X = 16Y + 52, we can rewrite it as:
X - 52 = 16Y
Step 2: Substitute this expression for the rewritten equation into the other equation:
X = 13Y - 52
X - 52 = 16Y
Step 3: Set the two expressions equal to each other:
16Y = 13Y - 52
Step 4: Simplify and solve for Y:
16Y - 13Y = -52
3Y = -52
Y = -52/3
The number of cakes sold Y is -52/3.
Step 5: Substitute the value of Y back into one of the original equations to solve for X:
X = 16Y + 52
X = 16(-52/3) + 52
X = -832/3 + 52
X = -832/3 + 156/3
X = -676/3
The sales and expenditures X are -676/3.
Therefore, the sales and expenditures will be approximately -$225.33 and the number of cakes sold will be approximately -17.33.