Problem 18 - Precalculus: Mathematical for Calculus

Question

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} x+y+z &=4 \\ -x+2 y+3 z &=17 \\ 2 x-y &=-7 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The unique solution is \( x = -2 \), \( y = 3 \), \( z = 3 \).

Step 1: Write the augmented matrix

First, we convert the given system of linear equations into an augmented matrix. The system is: 1. \( x + y + z = 4 \) 2. \( -x + 2y + 3z = 17 \) 3. \( 2x - y = -7 \) So, the augmented matrix is: \[\begin{bmatrix}1 & 1 & 1 & | & 4 \-1 & 2 & 3 & | & 17 \2 & -1 & 0 & | & -7 \end{bmatrix}\]

Step 2: Use Gaussian elimination to form an upper triangular matrix

The goal is to use row operations to make zeros below the pivots in the first column. Start with making the second row's first element zero by adding the first row to the second row: \[\begin{bmatrix}1 & 1 & 1 & | & 4 \0 & 3 & 4 & | & 21 \2 & -1 & 0 & | & -7 \end{bmatrix}\]Next, eliminate the first element in the third row by subtracting twice the first row from the third row:\[\begin{bmatrix}1 & 1 & 1 & | & 4 \0 & 3 & 4 & | & 21 \0 & -3 & -2 & | & -15 \end{bmatrix}\]

Step 3: Replace third row to achieve upper triangular form

To eliminate the second element in the third row, add the second row to the third row:\[\begin{bmatrix}1 & 1 & 1 & | & 4 \0 & 3 & 4 & | & 21 \0 & 0 & 2 & | & 6 \end{bmatrix}\]

Step 4: Solve for the variables

With the matrix in an upper triangular form, we can solve for the variables starting from the bottom row upwards:1. From row 3: \( 2z = 6 \Rightarrow z = 3 \)2. Substitute \( z = 3 \) into row 2: \( 3y + 4(3) = 21 \Rightarrow 3y = 9 \Rightarrow y = 3 \)3. Substitute \( y = 3 \) and \( z = 3 \) into row 1: \( x + 3 + 3 = 4 \Rightarrow x = -2 \)

Step 5: Verify the solution

Verify the solution \( x = -2, y = 3, z = 3 \) by substituting back into the original equations:- Equation 1: \( x + y + z = -2 + 3 + 3 = 4 \)- Equation 2: \( -x + 2y + 3z = 2 + 6 + 9 = 17 \)- Equation 3: \( 2x - y = -4 - 3 = -7 \)All original equations hold true, confirming the solution.

Key Concepts

Augmented Matrix

When dealing with a system of linear equations, an augmented matrix is a very useful tool. It acts as a condensed version of the system where both the coefficients and constants are included.
In our scenario, we have:

• Three equations
• Three variables ( {x, y, z})

The augmented matrix takes the coefficients from each of the equations and aligns them into rows. It also includes the constants on the right side, separated by a vertical line. This method enables you to work with the equations more easily, transforming them into a simplified structure. By doing this, you can perform operations that will allow you to find the value of each variable step by step. Remember, this step is crucial as it simplifies handling the system during operations like Gaussian elimination.

Upper Triangular Matrix

An upper triangular matrix is an important step using the process of Gaussian elimination. This form means that all the elements below the main diagonal are zero. Creating this form is essential because it simplifies solving the equations.
Take a look at our example:

• The first major diagonal has non-zero values.
• Everything below it, however, is zero.

With this formation, you can easily solve for one variable at a time, starting from the bottom row and working your way up. This process is known as back-substitution. Obtaining this upper triangular matrix often requires multiple steps of row operations, such as swapping, adding, or multiplying rows. These operations aim to "zero out" specific values so you can easily identify a path to the unique solution.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. In simpler terms, it is a collection of straight-line equations that all intersect at one solution point for the variables.
The major goal when dealing with such systems is to find this exact solution point ( x, y, z).

• Unique solutions imply intersection at one point.
• Infinite solutions mean the lines overlap entirely.
• No solution points to parallel lines that never meet.

In our exercise, we have three equations with three variables, and by following through Gaussian elimination, the system was reduced to an upper triangular form, which allowed us to systematically find the solution. Understanding how each row represents an equation while performing operations is crucial to solve these systems accurately.