Problem 51 - 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. Th ee numbers add up to \(106 .\) The fi st number is 3 less than the second number. The third number is 4 more than the fi st number.

Step-by-Step Solution

Answer

Yes, the unique solution is \( x = 33 \), \( y = 36 \), \( z = 37 \).

Step 1: Define Variables

Let's denote the first number as \( x \), the second number as \( y \), and the third number as \( z \).

Step 2: Set Up Equations

Using the information given, we can set up the following system of equations:1. \( x + y + z = 106 \) (The sum of the numbers is 106)2. \( x = y - 3 \) (The first number is 3 less than the second number)3. \( z = x + 4 \) (The third number is 4 more than the first number).

Step 3: Substitute to Simplify Equations

Substitute the expression for \( x \) from equation 2 and the expression for \( z \) from equation 3 into equation 1:\( (y - 3) + y + ((y - 3) + 4) = 106 \).

Step 4: Simplify and Solve for y

First, simplify the equation:\( y - 3 + y + y - 3 + 4 = 106 \).This simplifies to:\( 3y - 2 = 106 \).Add 2 to both sides:\( 3y = 108 \).Divide by 3:\( y = 36 \).

Step 5: Find x

Substitute \( y = 36 \) back into equation 2:\( x = 36 - 3 \).So, \( x = 33 \).

Step 6: Find z

Substitute \( x = 33 \) back into equation 3:\( z = 33 + 4 \).So, \( z = 37 \).

Step 7: Calculate the Determinant

The system of equations in matrix form is:\[\begin{bmatrix}1 & 1 & 1 \1 & -1 & 0 \1 & 0 & -1 \end{bmatrix}\begin{bmatrix}x \y \z\end{bmatrix}=\begin{bmatrix}106 \3 \-4\end{bmatrix}\]The determinant of the coefficient matrix is calculated as:\[ ext{det} = 1(-1)(-1) + 1(0)(1) + 1(1)(0) - [1(1)(0) + 1(0)(-1) + 1(-1)(1)] = 1 + 0 + 0 - (0 + 0 - 1) = 2.\]

Step 8: Conclusion

The determinant is non-zero (2 ≠ 0), indicating that there is a unique solution. The unique solution is \( x = 33 \), \( y = 36 \), \( z = 37 \).

Key Concepts

Determinant Calculation

In linear algebra, the determinant of a matrix is a special number that can be calculated from its elements. It provides crucial information about the matrix. The determinant can tell us whether a system of linear equations has a unique solution, infinitely many solutions, or no solution at all.

In our exercise, the system of linear equations is represented by a matrix. The calculation involves a formula that considers all the elements in the matrix. For a 3x3 matrix:

• The determinant is calculated by taking the sum of the products of the diagonals and subtracting certain products.
• In our case, it was: \[ \text{det} = 1(-1)(-1) + 1(0)(1) + 1(1)(0) - [1(1)(0) + 1(0)(-1) + 1(-1)(1)] \]
• The value we obtained was 2.

If the determinant is non-zero, like in this example, it shows that our linear system has a unique solution.

Unique Solution

A unique solution in the context of a system of linear equations means that there is only one set of values for the variables that satisfy all the equations simultaneously.

When we solve our specific system, we get:

• First equation: \( x + y + z = 106 \)
• Second equation: \( x = y - 3 \)
• Third equation: \( z = x + 4 \)

These equations can be considered in a matrix form. Once the determinant of this matrix is calculated and found to be non-zero, it confirms a unique solution.

This unique solution is specifically the set of values that solves all equations exactly. In our example, this was determined to be:

• \( x = 33 \)
• \( y = 36 \)
• \( z = 37 \)

These values satisfy each equation in the system, reinforcing their uniqueness due to the non-zero determinant.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces (also called linear spaces), and linear mappings between these spaces. It is a powerful tool for modeling complex problems in fields such as computer science, physics, and engineering.

The key elements in linear algebra include:

• Matrices and determinants, which are used to solve systems of linear equations.
• Vector spaces which provide a framework for analyzing linear mappings.

In the context of our system of linear equations:

• We used a matrix to represent the system, where the matrix's determinant helps in determining the nature of solutions.
• The determinant's role was vital in deciding whether our system has a unique solution, infinitely many solutions, or no solution.

Thus, linear algebra provides both the language and the tools necessary to tackle such mathematical problems efficiently.