Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. At an ice cream store, a family ordered 1 banana split, paying... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. At an ice cream store, a family ordered 1 banana split, paying a total of $6 for their order. The next customer ordered 1 banana split and 1 hot fudge sundae and paid $9. How much does each item cost? A banana split costs $ and a hot fudge sundae costs $.
Understand the Problem
The question asks to set up and solve a system of equations based on the costs of a banana split and a hot fudge sundae at an ice cream store, using an augmented matrix method.
Answer
The cost of a banana split is $6, and the cost of a hot fudge sundae is $3.
Answer for screen readers
A banana split costs $6 and a hot fudge sundae costs $3.
Steps to Solve
- Define Variables Let:
- ( x ) = cost of a banana split
- ( y ) = cost of a hot fudge sundae
- Set Up Equations Based on the problem:
-
The first equation from the first family’s order is: $$ x = 6 $$
-
The second equation from the second customer’s order is: $$ x + y = 9 $$
-
Write Augmented Matrix From the equations, we can construct the augmented matrix: $$ \begin{bmatrix} 1 & 0 & | & 6 \ 1 & 1 & | & 9 \end{bmatrix} $$
-
Use Row Reduction Now, we perform row operations to solve for ( x ) and ( y ):
- Subtract the first row from the second: $$ R_2 = R_2 - R_1 $$ This gives: $$ \begin{bmatrix} 1 & 0 & | & 6 \ 0 & 1 & | & 3 \end{bmatrix} $$
- Extract Solutions From this matrix, we see:
- ( x = 6 )
- ( y = 3 )
A banana split costs $6 and a hot fudge sundae costs $3.
More Information
The prices of the items were found through a system of equations method. These kinds of problems are common in algebra to help understand real-world applications of linear equations.
Tips
- Confusing the equations: Make sure each variable corresponds correctly to the quantities described.
- Incorrect row operations when solving the augmented matrix: Double-check calculations when manipulating the matrix.
AI-generated content may contain errors. Please verify critical information
