Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
Ella is starting a business selling handmade necklaces. She has decided to invest an initial amount of $57 for advertising, and materials cost $7 for each necklace she makes. Ella can sell her creations for $26 per necklace. Once she makes and sells a certain number of necklaces, she will break even, with identical expenses and sales. How many necklaces would that take? What would the total expenses and sales be then?
initial amount of $57
expense = 57
materials cost $7 for each necklace
expense = 57 + 7x
sales = 26x
...
Let's define the variables in this problem:
Let "x" represent the number of necklaces that Ella makes and sells.
Now, we can create the system of equations:
1) The total expenses equation:
Total expenses = initial investment + (cost per necklace * number of necklaces)
Total expenses = $57 + ($7 * x)
Total expenses = 57 + 7x
2) The total sales equation:
Total sales = selling price per necklace * number of necklaces
Total sales = $26 * x
Total sales = 26x
Now, we use the method of substitution:
We know that at the break-even point, total expenses = total sales. So, we can set up the equation:
57 + 7x = 26x
Next, we solve for x:
57 = 26x - 7x
57 = 19x
x = 57/19
x = 3
Therefore, Ella would need to make and sell 3 necklaces in order to break even.
To find the total expenses and sales at that point, we substitute the value of x back into the equations:
Total expenses = 57 + 7x
Total expenses = 57 + 7(3)
Total expenses = 57 + 21
Total expenses = $78
Total sales = 26x
Total sales = 26(3)
Total sales = 78
So, at the break-even point, Ella would have a total expense of $78 and total sales of $78.
Let's first define the variables in the problem:
Let "x" represent the number of necklaces Ella makes and sells.
Let "E" represent Ella's total expenses.
Let "S" represent Ella's total sales.
Now, let's write the equations to describe the situation:
1) Ella's total expenses (E) can be calculated by adding the initial advertising cost ($57) to the cost of materials for each necklace she makes ($7):
E = 57 + 7x
2) Ella's total sales (S) can be calculated by multiplying the selling price per necklace ($26) by the number of necklaces she makes and sells (x):
S = 26x
According to the problem, when Ella breaks even, her expenses equal her sales. So we can set the two equations equal to each other:
57 + 7x = 26x
Now, let's solve the equation using substitution to find the value of "x" and then calculate the total expenses and sales.
Subtract 7x from both sides of the equation:
57 = 19x
Divide both sides of the equation by 19:
x = 57/19
x = 3
Therefore, Ella would break even after making and selling 3 necklaces.
To find the total expenses (E) and total sales (S) at that point, substitute the value of x into the respective equations:
E = 57 + 7x
E = 57 + 7(3)
E = 57 + 21
E = 78
S = 26x
S = 26(3)
S = 78
Thus, when Ella sells 3 necklaces, her total expenses and sales would be $78 each.