Problem 51 - 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 up to 106 . The first number is 3 less than the second number. The third number is 4 more than the first number.

Step-by-Step Solution

Answer

The system has a unique solution with numbers: 33, 36, and 37.

Step 1: Define Variables

Let's define the variables for the three unknown numbers. Let \( x \) represent the first number, \( y \) represent the second number, and \( z \) represent the third number.

Step 2: Translate Conditions into Equations

Based on the problem statement, we can form the following system of equations: 1. The sum of the three numbers is 106: \( x + y + z = 106 \)2. The first number is 3 less than the second number: \( x = y - 3 \)3. The third number is 4 more than the first number: \( z = x + 4 \).

Step 3: Rearrange and Substitution

We will substitute equations 2 and 3 into equation 1 to get everything in terms of \( y \).From equation 2: \( x = y - 3 \).From equation 3: \( z = (y - 3) + 4 = y + 1 \).Substitute into equation 1:\( (y - 3) + y + (y + 1) = 106 \).

Step 4: Simplify the Equation

Simplify the equation obtained from Step 3:\( y - 3 + y + y + 1 = 106 \)\( 3y - 2 = 106 \)Add 2 to both sides:\( 3y = 108 \).Divide by 3:\( y = 36 \).

Step 5: Find Other Variables

Use the value of \( y \) to find \( x \) and \( z \):For \( x \):\( x = y - 3 = 36 - 3 = 33 \).For \( z \):\( z = x + 4 = 33 + 4 = 37 \).

Step 6: Form System of Equations Matrix

The system of equations \( x + y + z = 106 \), \( x - y = -3 \), and \( x - z = -4 \) can be written in matrix form as:\[\begin{bmatrix} 1 & 1 & 1 \ 1 & -1 & 0 \ 1 & 0 & -1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 106 \ -3 \ -4 \end{bmatrix} \].

Step 7: Calculate Determinant

The determinant of the coefficient matrix is:\[\text{det} = \begin{vmatrix} 1 & 1 & 1 \ 1 & -1 & 0 \ 1 & 0 & -1 \end{vmatrix} = 1(-1)(-1) + 1(0)(1) + 1(1)(0) - 1(1)(1) - 1(0)(-1) - 1(-1)(0) = 1 + 0 + 0 - 1 + 0 + 0 = 2\].

Step 8: Conclusion on Unique Solution

Since the determinant is \( 2 \), which is not zero, the system of equations has a unique solution. The unique solution is \( x = 33 \), \( y = 36 \), \( z = 37 \).

Key Concepts

Determinant Calculation

In linear algebra, the determinant is a special number that can be calculated from a square matrix. Understanding how to calculate the determinant helps us understand whether a system of equations has a unique solution.
For a given matrix, the determinant gives insight into various properties of the matrix.

• If the determinant is zero, the system of equations does not have a unique solution; it could have infinitely many solutions or none at all because the system is dependent or inconsistent.
• If the determinant is non-zero, the system generally has a unique solution, meaning the lines intersect at a single point.

In our example, after transforming the problem into a matrix and calculating the determinant of \[\begin{vmatrix} 1 & 1 & 1 \ 1 & -1 & 0 \ 1 & 0 & -1 \end{vmatrix}\],we find the determinant as 2. This non-zero value indicates that, indeed, there is a unique solution for the system of equations described.

Unique Solution

A unique solution in a system of linear equations means there is precisely one set of variable values that satisfy all the equations simultaneously. This is an important concept because it assures us that our solution is definitive and distinct.
When examining whether a system has a unique solution, the determinant is crucial. A non-zero determinant, as seen in this case with a value of 2, is a clear indication of the system having one solution. In our example, by solving the equations given, we determined that the numbers that satisfy the conditions are \( x = 33 \), \( y = 36 \), and \( z = 37 \). This outcome shows that each condition (equation) intersects precisely at one point, verifying the claim of a unique solution.

Matrix Form

Converting a system of equations into its matrix form is a helpful step, making it easier to compute and visualize complex systems. In matrix form, our example's system of equations became:\[\begin{bmatrix} 1 & 1 & 1 \ 1 & -1 & 0 \ 1 & 0 & -1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 106 \ -3 \ -4 \end{bmatrix}\]This expression consists of:

• The coefficient matrix, where the rows correspond to the coefficients of variables from each equation.
• The variable column matrix containing the unknowns \( x, y, z \).
• The constant column matrix showing the sums from each equation.

Matrix form aids in quickly applying various mathematical techniques such as determinant calculation and methods of solving systems, like Gaussian elimination or Cramer's rule, to effectively find solutions.

Algebraic Substitution

Algebraic substitution involves replacing one variable in an equation with an expression from another equation. It is a traditional approach that simplifies solving systems of equations.Let's explore this with our example:

• From one condition, we had \( x = y - 3 \) and used it to express \( x \) in terms of \( y \).
• In another condition, \( z = x + 4 \), we substituted the expression for \( x \) we found into this equation, leading to \( z = y + 1 \).
• Lastly, putting all these expressions together into the original sum equation resulted in a single equation in terms of \( y \): \( 3y - 2 = 106 \).

By solving this step-by-step, we were able to determine the exact values for all variables. Algebraic substitution reduced the problem to simpler terms, making it easier to isolate and determine the value of each variable in the system.