Problem 59 - College Algebra

Question

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?

Step-by-Step Solution

Answer

Tom ate 6, Joe ate 3, and Albert ate 3 ice cream bars.

Step 1: Define Variables

Let \( t \) represent the number of ice cream bars Tom ate, \( j \) represent the number of ice cream bars Joe ate, and \( a \) represent the number of ice cream bars Albert ate.

Step 2: Write the Equations

Based on the problem, we have the following equations: \( t = 2j \) (Tom ate twice as many as Joe), \( a = t - 3 \) (Albert ate three less than Tom), and the total equation \( t + j + a = 12 \) (the total number of ice cream bars is 12).

Step 3: Form the System of Equations

Substitute \( t = 2j \) and \( a = t - 3 \) in the total equation: \( 2j + j + (2j - 3) = 12 \). Simplifying this gives us \( 5j - 3 = 12 \).

Step 4: Solve for One Variable

Solve \( 5j - 3 = 12 \) to find \( j \): Add 3 to both sides to get \( 5j = 15 \), and then divide by 5 to find \( j = 3 \).

Step 5: Find the Value of \( t \)

Since \( t = 2j \), substitute \( j = 3 \) to find \( t = 2 \times 3 = 6 \).

Step 6: Find the Value of \( a \)

Use \( a = t - 3 \), substitute \( t = 6 \) to find \( a = 6 - 3 = 3 \).

Step 7: Check the Solution

Verify the total ice cream bars: \( t + j + a = 6 + 3 + 3 = 12 \), which equals the total in the problem.

Key Concepts

Matrix Inverse

The matrix inverse is a powerful tool used to solve systems of equations. Imagine it as the "opposite" or "undo" operation for a matrix, much like division is for multiplication. If a matrix is denoted as a square array of numbers, finding its inverse allows us to solve equations in which that matrix represents coefficients.
To determine an inverse, the matrix must be square (same number of rows as columns) and must have a non-zero determinant. If the determinant is zero, the matrix does not have an inverse, and the system cannot be solved using this method.
When we write a system of equations in matrix form, represented as \( AX = B \), the inverse helps us find \( X \), as \( A^{-1}AX = A^{-1}B \). This simplifies to \( X = A^{-1}B \). It's a neat algebra trick that transforms complex systems into simpler calculations, making it easier to find solutions.

Algebra

Algebra is the foundation of solving equations and inequalities through the manipulation and balancing of expressions. It involves a series of operations like addition, subtraction, multiplication, and division.
In this exercise, algebra helps us express the relationship between the number of ice cream bars each roommate eats. By defining variables according to these relationships, we create an algebraic system of equations.
For example, we defined that if Tom ate twice as many bars as Joe, the equation \( t = 2j \) is formed. Similarly, Tom eating three more than Albert leads to \( a = t - 3 \). These equations help us model the situation and eventually solve it using algebraic methods. By simplifying and solving these equations, we find the values of unknowns, making algebra an essential part of problem solving.

Problem Solving

Problem solving is an essential skill that involves analyzing a situation, identifying the problem, and finding solutions. In the context of a math problem like this, it requires understanding the relationships and constraints given in the scenario.
To successfully solve, you should:

• Identify what needs to be found, in this case, the number of ice cream bars each person ate.
• Understand the conditions and relationships described, such as Tom eating twice as many bars as Joe.
• Translate these relationships into mathematical equations.
• Apply appropriate methods, like using a matrix inverse or algebraic manipulation, to find the solution.

Problem solving can seem daunting, but breaking it down into steps makes it manageable. Think clearly about the information you have and the questions asked, and the path to the solution often becomes apparent.

Variable Substitution

Variable substitution is a technique used when solving equations by replacing one variable with another expression or value. It simplifies systems and makes it easier to handle each equation step-by-step.
In this exercise, substitution was key. We substituted the expressions for \( t \) and \( a \) into the total equation, simplifying three equations into one.
For example, starting with \( t = 2j \) and \( a = t - 3 \), when substituted into the total equation \( t + j + a = 12 \), the expressions are combined into a single equation in terms of \( j \): \( 5j - 3 = 12 \).
This allows us to solve for \( j \) directly, and then use the relationships previously defined to find \( t \) and \( a \). Substitution is like a shortcut, keeping calculations neat and preventing errors that could arise from juggling multiple variables at once.