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Demonstration of the substitution method is required for this assessment. Show all work!
The members of a cooking club are making cakes, which they will sell at a street fair for $16 apiece. It cost $52 for a booth at the fair, and the ingredients for each cake cost $3. At some point, the club members will sell enough cakes so that their sales cover their expenditures. How much will the sales and expenditures be? How many cakes will they have sold? Y and X slove for
Bot GPT 3.5
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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.