Jillian is selling boxes of cookies to raise money for her basketball team. The 10 oz box costs $3.50, while the 16 oz box costs $5.00. At the end of one week, she collected $97.50, selling a total of 24 boxes. The system of equations that models her sales is below.
[tex]\begin{array}{l}
x+y=24 \\
3.50 x+5.00 y=97.50
\end{array}[/tex]
Solve the system of equations. How many 10 oz boxes were sold?
Answer (1)
Express y in terms of x using the first equation: y = 24 − x .
Substitute this expression into the second equation: 3.50 x + 5.00 ( 24 − x ) = 97.50 .
Simplify and solve for x : − 1.50 x = − 22.50 , which gives x = 15 .
Therefore, the number of 10 oz boxes sold is 15 .
Explanation
Analyze the problem Let's analyze the problem. We have a system of two equations with two variables, x and y , representing the number of 10 oz and 16 oz boxes sold, respectively. The equations are:
x + y = 24 3.50 x + 5.00 y = 97.50
Our goal is to find the value of x , which represents the number of 10 oz boxes sold.
Express y in terms of x We can solve this system of equations using substitution or elimination. Let's use substitution. From the first equation, we can express y in terms of x :
y = 24 − x
Substitute into the second equation Now, substitute this expression for y into the second equation:
3.50 x + 5.00 ( 24 − x ) = 97.50
Simplify the equation Next, simplify and solve for x :
3.50 x + 120 − 5.00 x = 97.50 − 1.50 x = 97.50 − 120 − 1.50 x = − 22.50
Solve for x Now, divide both sides by -1.50 to find x :
x = − 1.50 − 22.50 x = 15
State the answer So, Jillian sold 15 boxes of 10 oz cookies.
Examples
Imagine you're organizing a bake sale for your school. You need to figure out how many of each type of treat to bake to reach your fundraising goal. This problem is similar to figuring out how many boxes of each size cookie Jillian sold. By setting up a system of equations, you can determine the exact number of each item needed to meet your target, ensuring you maximize your profits and minimize waste. This is a practical application of algebra in everyday scenarios.
