Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Three numbers add to 216 . The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.

Step-by-Step Solution

Answer

Yes, there's a unique solution: \( x=56, y=56, z=104 \).

Step 1: Define Variables

Let's assign variables to the numbers. Let \( x \), \( y \), and \( z \) be the three numbers. We have to find the values of \( x \), \( y \), and \( z \).

Step 2: Formulate the Equations

Based on the provided conditions, translate the word problems into equations:1. The sum of three numbers is 216: \( x + y + z = 216 \).2. The sum of the first two numbers is 112: \( x + y = 112 \).3. The third number is 8 less than the sum of the first two: \( z = x + y - 8 \).

Step 3: Simplify the Equations

Using the second equation \( x + y = 112 \), substitute \( z = 112 - 8 \) into the first equation. Therefore, \( z = 104 \).

Step 4: Substitute and Simplify the First Equation

Substitute \( z = 104 \) into \( x + y + z = 216 \):\( x + y + 104 = 216 \).Simplify to find \( x + y = 112 \), which is consistent with our second original equation.

Step 5: Solve for the First Two Variables

Given \( x + y = 112 \) and \( z = 104 \), substitute \( z \) back to find specific values for \( x \) and \( y \): Use consistent numbers to confirm that any two values satisfying \( x + y = 112 \) and an already given \( z \).

Step 6: Determine Unique Solution

To see if there's a unique solution, calculate the determinant of the coefficient matrix:The system can be expressed as:\[\begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 216 \ 112 \ 104 \end{bmatrix}\]The determinant of the matrix is \( 1(1 \cdot 1 - 0 \cdot 0) - 1(0 \cdot 1 - 0 \cdot 1) + 1(0 \cdot 1 - 1 \cdot 0) = 1 \); since it's not zero, a unique solution exists.

Step 7: Solution of the System

Given \( x + y = 112 \) and \( z = 104 \), selecting \( x = 56, y = 56 \) satisfies both initial conditions while maintaining consistency as:\( 56 + 56 = 112 \) and \( x + y + z = 216 \).

Key Concepts

Determinant

The determinant is a key concept in linear algebra, especially when dealing with systems of linear equations. It helps us determine certain properties of a matrix, such as whether a unique solution exists for the system. In simpler terms, the determinant is a special number that can be calculated from a square matrix. It acts like a diagnostic tool for matrices.

For our exercise, the determinant is calculated from the coefficient matrix:

• Write down the given matrix: \[ \begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]
• Compute the determinant using the method of expansion along any row or column: \[ 1(1 \cdot 1 - 0 \cdot 0) - 1(0 \cdot 1 - 0 \cdot 1) + 1(0 \cdot 1 - 1 \cdot 0) \]
• The result is 1, which is not zero.

A non-zero determinant indicates certain properties about the system of linear equations: it confirms that we can find a unique solution for the variables in our equations.

Unique Solution

A unique solution in a system of linear equations means there is exactly one set of values for the variables that satisfies all the equations. For this to happen, the determinant of the coefficient matrix must be non-zero. This indicates the equations are independent and can be solved to find a single answer.

In the given exercise, the calculated determinant is 1, which is clearly not zero. This guarantees that the system of equations has a unique solution. So, when we calculate and find specific numbers for our variables, we can be confident that they are indeed the only correct ones, provided the system is consistent.

Thus, if you have correctly set up your equations and determinant is non-zero, solving the system using substitution, elimination, or matrix operations will give you the unique solution.

Matrix Representation

Using matrix representation is a powerful way to organize and solve systems of linear equations. Instead of juggling multiple equations, you can use matrices to handle them systematically.

In our problem, we represent the system of equations as a matrix equation:

• The coefficient matrix: \[ \begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]
• The variable matrix: \[ \begin{bmatrix} x \ y \ z \end{bmatrix} \]
• The constant matrix: \[ \begin{bmatrix} 216 \ 112 \ 104 \end{bmatrix} \]

These matrices form the entire system as one matrix equation \( A\mathbf{x} = \mathbf{b} \). With this setup, solving the system becomes a matter of using matrix operations.

Using matrix algebra to solve for \( \mathbf{x} \) can involve finding the inverse of the coefficient matrix or using methods like row reduction. This representation not only makes solving easier but also reveals insights, such as whether the system has a unique solution, by checking conditions like the determinant.

Variable Definition

Defining variables is the foundational step in solving systems of linear equations. It involves assigning simple, clear representations to the unknowns you need to solve for. This step frames the entire problem and ensures you can translate word problems into precise mathematical equations.

In this exercise, the variables are defined as:

• \( x \) for the first number
• \( y \) for the second number
• \( z \) for the third number

Defining the variables simplifies setting up the equations from the given problem statement. For example:

• The statement 'three numbers add to 216' becomes \( x + y + z = 216 \).
• 'The sum of the first two numbers is 112' becomes \( x + y = 112 \).
• 'The third number is 8 less than the first two numbers combined' becomes \( z = x + y - 8 \).

Clear and thoughtful variable definitions lead to accurate equations, which are vital for successfully solving the system. It helps in devising strategies to find their values and in understanding the problem's logic.