Problem 49 - 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 \(56 .\) One number is 20 less than the other.

Step-by-Step Solution

Answer

The unique solution is \( x = 18 \) and \( y = 38 \).

Step 1: Define Variables

To begin solving the problem, let's define our variables. Let \( x \) be the first number and \( y \) be the second number.

Step 2: Set Up Equations

Based on the problem statement, we can set up two equations:1. The sum of the two numbers is given by the equation \( x + y = 56 \).2. One number is 20 less than the other, described by the equation \( x = y - 20 \).

Step 3: Form the System of Equations

The system of equations is:\[ \begin{align*} x + y &= 56, \ x &= y - 20. \end{align*} \]

Step 4: Write in Matrix Form

Express the system as a matrix equation:\[ \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 56 \ -20 \end{pmatrix} \]

Step 5: Calculate the Determinant

The determinant for the matrix \( A = \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \) is calculated as follows:\[ \text{det}(A) = (1)(-1) - (1)(1) = -1 - 1 = -2 \]

Step 6: Determine Unique Solution

Since the determinant is non-zero (\(-2\)), there is a unique solution to the system of equations.

Step 7: Solve the System of Equations

Substitute \( x = y - 20 \) into the first equation: \[ y - 20 + y = 56 \]Simplify: \[ 2y - 20 = 56 \]Add 20 to both sides: \[ 2y = 76 \]Divide by 2: \[ y = 38 \]Substitute \( y = 38 \) back to find \( x \): \[ x = 38 - 20 = 18 \]

Step 8: Verify the Solution

Check the original equations:1. \( x + y = 18 + 38 = 56 \) ✔️2. \( x = y - 20 = 38 - 20 = 18 \) ✔️Both equations are satisfied with \( x = 18 \) and \( y = 38 \).

Key Concepts

Determinant Calculation

The determinant is a critical value that helps to determine whether a system of linear equations has a unique solution. To find it, we look at the coefficient matrix of our system of equations. In this particular problem, our system is represented by the matrix:

• \( \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \)

To calculate the determinant of this matrix, we use the formula for a 2x2 matrix:\[ \text{det}(A) = a_{11} \cdot a_{22} - a_{12} \cdot a_{21} \]In our case, substituting the values from the matrix gives:

• \( \text{det}(A) = (1)(-1) - (1)(1) \)
• = -1 - 1 \
• = -2

The determinant of \(-2\) indicates that the matrix is invertible, implying that the system of linear equations has a unique solution.

Unique Solution

A unique solution occurs when a system of linear equations intersects at exactly one point in the solution space. For this to happen, the determinant of the coefficient matrix must be non-zero. For our system of equations:

• \( x + y = 56 \)
• \( x = y - 20 \)

After forming the matrix\( \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \),we calculated its determinant as \(-2\). Since this value is non-zero, it confirms the existence of a unique solution. When we solve the system of equations

• \(y - 20 + y = 56 \)
• \(2y = 76 \)
• \(y = 38 \)

we use these results to find the corresponding value of \(x\):

• \(x = y - 20 = 18\)

Therefore, the unique solution to this system is \(x = 18\) and \(y = 38\).

Matrix Representation

Matrix representation is a powerful method used to express a system of equations. It allows us to solve systems efficiently using computational techniques. For the given problem, where two numbers satisfy the system:

• \( x + y = 56 \)
• \( x = y - 20 \)

We can represent this system in matrix form as \( A \cdot \vec{x} = \vec{b} \):\[ \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 56 \ -20 \end{pmatrix} \]This succinct form provides a compact visualization of the operation being performed on the variables \(x\) and \(y\). Here:

• \( A \) is the coefficient matrix \( \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \)
• \( \vec{x} \) is the column matrix of variables \( \begin{pmatrix} x \ y \end{pmatrix} \)
• \( \vec{b} \) is the constant column matrix \( \begin{pmatrix} 56 \ -20 \end{pmatrix} \)

Matrix representation simplifies the process of solving systems using linear algebra techniques like determinant calculation and matrix inversion, ultimately helping us quickly and accurately find the solution to the equations.