Problem 50 - 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. Two numbers add up to 104 . If you add two times the first number plus two times the second number, your total is 208

Step-by-Step Solution

Answer

No unique solution; the determinant is zero.

Step 1: Define Variables

Let's define the variables: Let the first number be \( x \) and the second number be \( y \).

Step 2: Create System of Equations

From the problem statement, we have two equations: 1. The sum of the two numbers is 104: \[ x + y = 104 \]2. Two times the sum of the two numbers is 208: \[ 2x + 2y = 208 \]

Step 3: Simplify the Second Equation

Since both terms in the second equation are multiplied by 2, we can simplify it by dividing throughout by 2: \[ x + y = 104 \].

Step 4: Analyze and Solve the System

Notice that both equations simplify to the same expression, \( x + y = 104 \). This means we have infinitely many solutions as both equations are essentially the same, not two independent equations.

Step 5: Calculate the Determinant

To determine if there is a unique solution by checking the determinant, set the equations in standard form: \[\begin{align*}x + y &= 104 \2x + 2y &= 208\end{align*}\]The matrix form for this system using coefficients is: \[ \begin{bmatrix} 1 & 1 \ 2 & 2 \end{bmatrix} \].Calculate the determinant: \[ ext{det} = (1)(2) - (1)(2) = 0 \].Since the determinant is 0, the system does not have a unique solution.

Key Concepts

Determinant

The determinant, in the realm of linear algebra, is a special scalar value calculated from a square matrix. It gives us valuable insight into the system of equations represented by this matrix. In simple terms, the determinant helps us determine whether a system of linear equations has a unique solution. When we look at a system in matrix form, if the determinant of the matrix is zero, the system doesn't have a unique solution. This is because a zero determinant indicates that the rows of the matrix are linearly dependent. For our given problem, the matrix of coefficients is \(\begin{bmatrix} 1 & 1 \ 2 & 2 \end{bmatrix}\). The determinant here is \((1)(2) - (1)(2) = 0\), highlighting that there is no unique solution for this system.

Infinite Solutions

Infinite solutions occur when there is not just one specific answer to a set of linear equations. This typically happens if all the equations in the system essentially describe the same line or plane, which overlap completely. In the example exercise, we derived two equations: \(x + y = 104\) and \(2x + 2y = 208\). Upon simplifying, they both reduce to the same equation: \(x + y = 104\). Both equations are fully equivalent, leading to infinitely many solutions, as any value for \(x\) or \(y\) that satisfies the equation \(x + y = 104\) works for both equations, effectively describing a whole line of solutions.

Linear Algebra

Linear algebra helps us solve problems involving linear equations and their representations in a matrix form. It is the framework used in this exercise to understand the relationship between variables and equations. Fundamentally, linear algebra deals with concepts like vectors, matrices, and determinants to find solutions to systems of linear equations. By transforming equations into matrices, we can inspect various properties, such as matrix rank or determinants, to gain insights into system solutions. In this particular exercise, linear algebra allowed us to form a matrix based on the equations and to then analyze the determinant to find the number of solutions.

Solution Uniqueness

The uniqueness of a solution in a system of linear equations refers to whether there is one, single solution or not. When dealing with systems of equations, determining the determinant is crucial. A non-zero determinant indicates that the system has a unique solution, meaning that the equations intersect at precisely one point. In contrast, our exercise resulted in a zero determinant, indicating no unique solution was possible due to both equations describing the same constraint. Thus, the equations in our system are not linearly independent, meaning the system has an infinite number of solutions, not just a single unique one as they lie on the same line.