Problem 50 - College Algebra

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 fi st number plus two times the second number, your total is 208

Step-by-Step Solution

Answer

Infinite solutions, not unique.

Step 1: Define the Variables

Let \( x \) be the first number and \( y \) be the second number. This will allow us to translate the word problem into a mathematical system of equations.

Step 2: Write the System of Equations

Based on the problem, we have two equations. First, the two numbers add up to 104: \( x + y = 104 \). Second, two times the first number plus two times the second number equals 208: \( 2x + 2y = 208 \).

Step 3: Simplify the Second Equation

Notice that the second equation, \( 2x + 2y = 208 \), can be simplified by dividing through by 2, giving us \( x + y = 104 \).

Step 4: Analyze the System

Both our equations end up being \( x + y = 104 \). This is actually not two independent equations but a single equation repeated, indicating an infinite number of solutions rather than a unique solution.

Step 5: Calculate the Determinant

In order to solve for the determinant of the system, use the coefficients of the variables: \( \begin{bmatrix} 1 & 1 \ 2 & 2 \end{bmatrix} \). The determinant is calculated by \( 1\times 2 - 1\times 2 = 0 \).

Step 6: Conclusion on Solutions

Since the determinant is 0, the system of equations does not have a unique solution. Instead, it implies either no solution or infinitely many solutions. In this scenario, because both equations are the same, we have infinitely many solutions.

Key Concepts

Determinant Calculation

The determinant is a key mathematical tool used to determine whether a system of linear equations has a unique solution. In our exercise, we formed a system of equations from the problem:

• \( x + y = 104 \)

• \( 2x + 2y = 208 \)

To find the determinant, we first simplify the second equation to the form \( x + y = 104 \), by dividing the entire equation by 2. Now, the original system has the coefficients matrix: \[\begin{bmatrix} 1 & 1 \ 2 & 2 \end{bmatrix}\] The determinant of this matrix is calculated as follows: \[1 \times 2 - 1 \times 2 = 0\] The resulting value of the determinant is 0. This zero determinant is crucial because it tells us there is no unique linear combination of these equations.

Infinite Solutions

Infinite solutions occur when the system of equations is essentially the same equation repeated, which provides an infinite number of solutions. In the exercise, after simplification, both equations resulted in:

• \( x + y = 104 \)

This indicates they represent the same linear equation. Consequently, they do not intersect at a single point, which is typical of a unique solution. Instead, every pair of \( x \) and \( y \) that adds up to 104 is a solution. This is why we have infinite solutions. For example:

• \( x = 50, y = 54 \)
• \( x = 20, y = 84 \)
• \( x = 104, y = 0 \)

Each of these is a valid solution, and there are many more.

Unique Solution

A unique solution in a system of linear equations exists when every equation in the system intersects at exactly one point. This means that there is only one set of values that satisfies all the equations. For a unique solution to exist, the determinant calculated for the coefficient matrix must not be zero. In our exercise, however, because the determinant is zero, it immediately tells us that a unique solution is impossible in this case. Had the determinant been a non-zero number, it would mean that the equations would intersect at precisely one coordinate pair, giving us a single, clear answer. But here, due to the overlap of equations (reflected in a zero determinant), the set of solutions extends infinitely.

Mathematical Translation

Translating word problems into mathematical equations is a fundamental skill. It involves carefully interpreting the problem's language to form equations that represent the given conditions. In our example, we had two statements:

• "Two numbers add up to 104," which translates to \( x + y = 104 \).
• "Two times the first number plus two times the second number is 208," which translates to \( 2x + 2y = 208 \).

The act of translating involves identifying the variables (often unknown from the word problem) and constructing relevant equations. It's important to simplify and manipulate these mathematical representations to fully understand the relationships and eventually solve the problem.

Solving Equations

Solving equations is the process of finding the values for the unknown variables that satisfy all the given equations in a system. Once equations are established from a problem and simplifications are made, it's essential to decide how to approach solving.In our case, since both equations boiled down to \( x + y = 104 \), it was clear that solving meant simply recognizing all the potential pairs of numbers that add up to the given total. If the system had been more complex with distinct equations, various methods such as substitution, elimination, or matrix solutions might be employed. But here, the recognition of a repeated equation quickly directed us to understand that all combinations satisfying this equation solve the system.