Problem 27 - Algebra and Trigonometry

Question

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l}{2 x-3 y-z=13} \\ {-x+2 y-5 z=6} \\ {5 x-y-z=49}\end{array}\right. $$

Step-by-Step Solution

Answer

The unique solution is \( x = 7, y = 8, z = 2 \).

Step 1: Write the System as an Augmented Matrix

Convert the given system of equations into an augmented matrix. The system is:\[\begin{align*} 2x - 3y - z &= 13, \ -x + 2y - 5z &= 6, \ 5x - y - z &= 49. \end{align*}\] The augmented matrix is:\[\begin{bmatrix} 2 & -3 & -1 & | & 13 \ -1 & 2 & -5 & | & 6 \ 5 & -1 & -1 & | & 49 \end{bmatrix}.\]

Step 2: Apply Row Operations to Achieve Upper Triangular Form

Use row operations to begin converting the augmented matrix into an upper triangular form. First, make the leading coefficient of the first row 1. Perform \( R_1 \) operations to achieve this, e.g., swap rows if necessary. Next, eliminate the \( x \) term from the second and third rows by replacing each with \( R_2 + \frac{1}{2}R_1 \) and \( R_3 - \frac{5}{2}R_1 \).

Step 3: Continue to Reduce to Row Echelon Form

Now focus on the second row to make the leading coefficient (other than those in the leading position) 1. Perform row operations as needed, often involving substitutions or multiplications. In this example, update \( R_2 \) as necessary, and use it to eliminate the second column entry from \( R_3 \) to equalize to 0 using something like \( R_3 - kR_2 \).

Step 4: Back Substitution for Gauss-Jordan Elimination

After converting to row-echelon form, make the entire diagonal, leading to identity matrix form by further row operations if using Gauss-Jordan. The result would be a diagonal of 1s top to bottom with all other terms zero, allowing you to directly read off \( x, y, \) and \( z \).

Step 5: Find Solution from Simplified Matrix

At the Gauss-Jordan elimination completion, the matrix is:\[\begin{bmatrix} 1 & 0 & 0 & | & x \ 0 & 1 & 0 & | & y \ 0 & 0 & 1 & | & z \end{bmatrix},\] where the right-hand side solutions correspond to \( x, y, \) and \( z \) directly.

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations working together to define how variables relate to each other. Imagine equations as a group of friends making plans for a party, where each equation provides information and constraints about what can happen. For example, suppose you have three equations:

• \(2x - 3y - z = 13\)
• \(-x + 2y - 5z = 6\)
• \(5x - y - z = 49\)

Each equation has variables \(x, y, \) and \(z\), representing unknown values we seek to discover. The goal when solving these equations together is to find a single set of values for \(x, y,\) and \(z\) that satisfy all these equations simultaneously. This is referred to as finding a 'unique solution'. If a solution is found where all the conditions set by each equation are met, then the problem is solved successfully.

Augmented Matrix

An augmented matrix turns our text-based system of equations into a numerical grid that is easier to manipulate. This grid captures all coefficients of variables and the constant terms from each equation, allowing us to perform mathematical operations efficiently. Here's how it looks for our example:

• Equations: \(2x - 3y - z = 13\), \(-x + 2y - 5z = 6\), \(5x - y - z = 49\).
• Augmented Matrix: \[\begin{bmatrix} 2 & -3 & -1 & | & 13 \ -1 & 2 & -5 & | & 6 \ 5 & -1 & -1 & | & 49 \end{bmatrix}\]

Think of this matrix as rearranging the system into rows and columns where every row corresponds to an equation and each column represents a particular variable or the constant on the right-hand side. The vertical bar separates the constant terms from the variable coefficients, helping us remember what comes from the left against what's fixed on the right in equations.

Row Operations

Row operations are calculated maneuvers changing the rows of an augmented matrix. These operations help simplify and eventually solve the system of linear equations. They include:

• Swapping rows: Switches the position of two rows, which can help place non-zero entries where needed.
• Multiplying a row: Changes the row's values to make calculations easier, like turning a leading coefficient to 1.
• Adding/subtracting rows: Allows combining rows to eliminate certain terms from equations, crucial for simplifying equations. For example, multiplying one row and adding it to another can help clear coefficients in a specific column.

Imagine turning a cluttered room into a neat space by reordering and combining objects. Row operations move, multiply, and combine numbers in a similar way to bring the matrix closer to a form we can easily work with.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an advanced method for solving systems of linear equations using an augmented matrix.It is a detailed process aiming to create a matrix where each row corresponds to a single variable solved. Here's how to use this technique effectively:

• Reduce matrix to row-echelon form: Use row operations to achieve an upper triangular matrix where each leading number in a row is 1, and zeros are below these leading 1s.
• Diagonalization to identity matrix: Further simplify until the matrix is diagonal with ones on the diagonal, mimicking an identity matrix, where all other elements become zero.
• This form clearly maps solutions, allowing you to instantly determine the values of \(x, y,\) and \(z\).

By pursuing Gauss-Jordan elimination, you convert a complex chain of equations into manageable, separate equations in a structured and logical manner. It turns a matrix into a form resembling a solved Sudoku puzzle, where each number indicates a solution directly mapping to one variable.