Problem 38 - Precalculus Mathematics for Calculus

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} y-5 z &=7 \\ 3 x+2 y &=12 \\ 3 x &+10 z=80 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is dependent with solution: \( x = -31.33, \ y = 53, \ z = 9.2 \).

Step 1: Write the System of Equations in Matrix Form

Convert the given system of equations to an augmented matrix. The system is:\[ \begin{align*} \begin{cases} y - 5z &= 7 \ 3x + 2y &= 12 \ 3x + 10z &= 80 \end{cases} \end{align*} \] The augmented matrix is: \[ \begin{bmatrix} 0 & 1 & -5 & | & 7 \ 3 & 2 & 0 & | & 12 \ 3 & 0 & 10 & | & 80 \end{bmatrix} \]

Step 2: Perform Row Reduction

Use Gaussian elimination to row-reduce the matrix to its row-echelon form.\[ \begin{bmatrix} 0 & 1 & -5 & | & 7 \ 3 & 2 & 0 & | & 12 \ 3 & 0 & 10 & | & 80 \end{bmatrix} \] Subtract Row 1 from Row 3 to get a new Row 3: \[ \begin{bmatrix} 0 & 1 & -5 & | & 7 \ 3 & 2 & 0 & | & 12 \ 0 & -2 & 10 & | & 68 \end{bmatrix} \]

Step 3: Simplify and Check for Consistency

Simplify the matrix further by adding twice Row 2 to Row 3: \[ \begin{bmatrix} 0 & 1 & -5 & | & 7 \ 3 & 2 & 0 & | & 12 \ 0 & 0 & 10 & | & 92 \end{bmatrix} \] Divide Row 3 by 10: \[ \begin{bmatrix} 0 & 1 & -5 & | & 7 \ 3 & 2 & 0 & | & 12 \ 0 & 0 & 1 & | & 9.2 \end{bmatrix} \] This matrix is consistent as all equations can be solved without contradiction.

Step 4: Express Solution in Terms of Free Variables

Identify the basic and free variables. Since we have pivot positions in columns corresponding to \(y\) and \(z\), and a free variable in \(x\), the solution for the system is expressed in terms of \(x\).\[ \begin{align*} z &= 9.2 \ y - 5(9.2) &= 7 \ 3x + 2y &= 12 \end{align*} \] Solve for \(y\): \[ y = 53 \ \] Substitute \(y = 53\) into the second equation: \[ 3x + 2(53) = 12 \] Solve for \(x\): \[ x = -31.33 \]

Step 5: Write the Complete Solution

The solution of the system is: \[ x = -31.33, \ y = 53, \ z = 9.2 \] Therefore, the system is dependent and has the solution given above.

Key Concepts

matrix row reduction

Matrix row reduction is a key technique used in solving systems of linear equations. It involves transforming a matrix into a simpler form, which can make the system easier to solve. This process often utilizes row operations which include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting the multiple of one row to another row.

In the context of our system, we perform row reduction to simplify the augmented matrix so that it gradually takes the form of what we call 'row-echelon form'. Through these manipulations, we aim to get zeros in the lower left-hand part of the matrix, ending with the last row simplified to isolate and solve for specific variables.

Gaussian elimination

Gaussian elimination is the process of systematically performing row operations on a matrix to achieve row-echelon form, and, if needed, further simplifying to reduced row-echelon form. It's one of the fundamental algorithms in linear algebra.

The objective of Gaussian elimination is to simplify the matrix to the extent where we can back-substitute to find the solutions of the variables involved. For our system, Gaussian elimination was utilized to reduce the augmented matrix, enabling the identification of dependent variables and offering insights into the structure of the solutions. This technique highlights how interconnected each equation is and works to pinpoint any redundancy or dependencies within the system.

dependent system

A dependent system indicates that at least one equation in the system is a linear combination of the others. This means that not all equations provide unique information about the variables.

• For instance, in our exercise, after simplifying the system, it was found consistent without contradiction.
• The solution is expressed in terms of free variables, indicating dependency.

The final result demonstrated that we could find a solution, confirming the dependent nature of the system. Essentially, the solution set was not unique, signifying the equations rely on each other, exhibiting redundancy within their mathematical relationships.

augmented matrix

The augmented matrix is an essential tool in representing a system of linear equations. It consists of the coefficients of the variables and an added column representing the constants from each equation.

In the given problem, the augmented matrix is constructed from the set of equations provided. By transforming this linear system into an augmented matrix, it becomes easier to apply row reduction methods like Gaussian elimination efficiently.

• This not only helps visualize the relationships between the equations but also simplifies processing through various linear operations.
• Ultimately, this matrix format was crucial in solving the system and identifying its dependency.