Problem 20 - Precalculus: Mathematical for Calculus

Question

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination.\ $$\left\\{\begin{aligned} 2 y+z &=4 \\ x+y &=4 \\ 3 x+3 y-z &=10 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The unique solution is \(x=3\), \(y=1\), and \(z=2\).

Step 1: Write the Augmented Matrix

First, represent the given system of equations in the form of an augmented matrix. The system is:\(\begin{aligned} 0x + 2y + 1z &= 4 \ 1x + 1y + 0z &= 4 \ 3x + 3y - 1z &= 10 \end{aligned}\). The corresponding augmented matrix is: \[ \begin{bmatrix} 0 & 2 & 1 & | & 4 \ 1 & 1 & 0 & | & 4 \ 3 & 3 & -1 & | & 10 \end{bmatrix} \]

Step 2: Swap Rows (if necessary)

We should have a non-zero element in the first column of the first row. Swap Row 1 with Row 2:\[ \begin{bmatrix} 1 & 1 & 0 & | & 4 \ 0 & 2 & 1 & | & 4 \ 3 & 3 & -1 & | & 10 \end{bmatrix} \]

Step 3: Eliminate First Column Below the Pivot

Use the first row to make the entry below the first pivot (3 in Row 3) zero:- Subtract 3 times Row 1 from Row 3: - New Row 3: \((3-3\times1)x + (3-3\times1)y - (1-3\times0)z = (10-3\times4)\)The matrix becomes:\[ \begin{bmatrix} 1 & 1 & 0 & | & 4 \ 0 & 2 & 1 & | & 4 \ 0 & 0 & -1 & | & -2 \end{bmatrix} \]

Step 4: Back Substitution and Obtain Row Echelon Form

Now perform back substitution to find the solutions for each variable:- From Row 3, solve for \(z\): \(-1z = -2 \Rightarrow z = 2\) - Substitute \(z = 2\) into Row 2: \(2y + 1\times2 = 4 \Rightarrow 2y = 2 \Rightarrow y = 1\)- Substitute \(y=1\) into Row 1: \(x + 1 = 4 \Rightarrow x = 3\)

Step 5: Final Solution Verification

Verify if the solution satisfies the original system:- Equation 1: \(2(1) + 2 = 4\)- Equation 2: \(3 + 1 = 4\)- Equation 3: \(3 \times 3 + 3 \times 1 - 2 = 10\)Since all equations are satisfied, the solution is correct.

Key Concepts

Augmented Matrix

When solving a system of linear equations, an augmented matrix is a key tool that simplifies the process of finding solutions. An augmented matrix is a rectangular array of numbers obtained from the coefficients and constants of the equations. In the given system:- Equation 1: \(0x + 2y + 1z = 4\)- Equation 2: \(1x + 1y + 0z = 4\)- Equation 3: \(3x + 3y - 1z = 10\)This system is represented in matrix form as:\[\begin{bmatrix}0 & 2 & 1 & | & 4 \1 & 1 & 0 & | & 4 \3 & 3 & -1 & | & 10\end{bmatrix}\]The vertical line in the augmented matrix separates the coefficients of the variables in the system from the constants on the right-hand side of each equation. By working with this matrix, we can apply row operations to simplify our work with the system.

Gauss-Jordan Elimination

Gauss-Jordan elimination is a method used to solve systems of linear equations by transforming the augmented matrix into a simpler form. The process involves a sequence of row operations to achieve an identity matrix on the left side, leading us directly to the solution of the system. Key steps in Gauss-Jordan elimination include:

• Swapping Rows: If there is a zero in a pivot position, row swapping helps place a non-zero number in that position.
• Row Multiplication: Multiply a row by a non-zero constant to change the pivot to 1.
• Row Addition/Subtraction: Add or subtract multiples of one row from another to introduce zeros below the pivot positions.

The end goal is to convert the matrix into reduced row echelon form, where each leading 1 in a row is the only non-zero entry in its column. This method not only simplifies the matrix but also directly provides the solutions to the variables.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations form the basis of linear algebra and are fundamental in understanding more complex algebraic structures.In the exercise, we have three linear equations involving three variables \(x\), \(y\), and \(z\):- Equation 1: \(0x + 2y + 1z = 4\)- Equation 2: \(1x + 1y + 0z = 4\)- Equation 3: \(3x + 3y - 1z = 10\)These equations can represent different geometrical concepts such as lines and planes in a three-dimensional space. Solving them collectively gives the intersection point, or a unique solution for each variable. This set of values will satisfy all the equations simultaneously.

Row Echelon Form

Row echelon form is a crucial step in the process of solving systems of linear equations through Gaussian elimination. In this form, the matrix has a staircase-like shape, with each leading entry of a row being to the right of the leading entry of the row above it.Characteristics of row echelon form are:

• All zero rows (if any) are at the bottom of the matrix.
• The leading entry of each non-zero row after the first appears to the right of the leading entry of the previous row.
• Leading entries are common referred to as pivots.

Once the matrix is in row echelon form, back substitution is used to find the values of the unknowns. This involves solving the equations from the last row up to the top. In the given exercise, we reached row echelon form and solved for \(z\), \(y\), and \(x\) accordingly, verifying the solution against the original system of equations. This step is crucial to ensure that the process has the correct and precise solution.