Problem 106 - Precalculus

Question

Solve the system of equations using an augmented matrix. $$ \begin{aligned} x+3 y+2 z &=4 \\ 3 x+10 y+9 z &=17 \\ 2 x+7 y+7 z &=17 \end{aligned} $$ Solution: $$ \begin{array}{c} \text { Step 1: Write the system as an } \\ \text { augmented matrix. } \end{array}\left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \\ 3 & 10 & 9 & 17 \\ 2 & 7 & 7 & 17 \end{array}\right] $$ \(\begin{aligned} \text { Step 2: Reduce the matrix using } &\left[\begin{array}{lll|r}1 & 0 & -7 & -11 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 4\end{array}\right] \\ \text { Gaussian elimination. } & 0 \end{aligned}\) Step 3: Identify the answer: \(\quad x=7 t-11\) Infinitely many solutions. \(\quad y=-3 t+5\) \(z=t\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The system is inconsistent; there is no solution.

Step 1: Set up the augmented matrix

Convert the system of equations into an augmented matrix format. This matrix will represent the system and allow us to apply Gaussian elimination:\[\left[ \begin{array}{ccc|c}1 & 3 & 2 & 4 \3 & 10 & 9 & 17 \2 & 7 & 7 & 17\end{array} \right]\]

Step 2: Apply row operations to create zeros below the first pivot

Use elementary row operations to create zeros below the first element (pivot) in the first column by modifying the rows:\[R_2 = R_2 - 3R_1 : \left[ \begin{array}{ccc|c}1 & 3 & 2 & 4 \0 & 1 & 3 & 5 \2 & 7 & 7 & 17\end{array} \right]\]\[R_3 = R_3 - 2R_1 : \left[ \begin{array}{ccc|c}1 & 3 & 2 & 4 \0 & 1 & 3 & 5 \0 & 1 & 3 & 9\end{array} \right]\]

Step 3: Continue to simplify the matrix

Further reduce by eliminating elements below the second pivot (second row, second column):\[R_3 = R_3 - R_2 : \left[ \begin{array}{ccc|c}1 & 3 & 2 & 4 \0 & 1 & 3 & 5 \0 & 0 & 0 & 4\end{array} \right]\]

Step 4: Analyze results of the reduced matrix

Observe that the last row translates to the equation: \(0x + 0y + 0z = 4\), which is equivalent to \(0 = 4\). This indicates a contradiction and suggests the system of equations is inconsistent.

Step 5: Conclude that there is no solution

Since an inconsistent system results from a contradiction, the solution is that there are no values for \(x\), \(y\), and \(z\) that satisfy all the equations simultaneously. Thus, the original system of equations has no solution.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously. For example, consider the system: \[\begin{aligned} x+3y+2z &= 4 \ 3x+10y+9z &= 17 \ 2x+7y+7z &= 17 \end{aligned} \] Our task is to identify values of \(x\), \(y\), and \(z\) that make all these equations true at the same time. Systems can be categorized as:

• Consistent: At least one solution exists.
• Inconsistent: No solutions exist.
• Dependent: Infinitely many solutions exist, typically when equations are multiples of each other.

For solving, we commonly use methods such as substitution, elimination, and this brings us to our next topic, Gaussian elimination.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves using a sequence of operations to transform a system into an easier form, often leading to what is known as "row-echelon form" or even "reduced row-echelon form." The key steps are:

• Form the augmented matrix from the system of equations.
• Use row operations to create zeros below each pivot (leading coefficient).
• Solve the resulting triangular system for the variables.

To illustrate, let's see the transformation from our initial example: Create the augmented matrix: \[ \left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \ 3 & 10 & 9 & 17 \ 2 & 7 & 7 & 17 \end{array}\right] \] Use operations like \(R_2 = R_2 - 3R_1\) to generate zeros under the first pivot: \[ \left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \ 0 & 1 & 3 & 5 \ 0 & 1 & 3 & 9 \end{array}\right] \] Continue eliminating below new pivots until the last row appears like \([0, 0, 0 | 4]\), suggesting further analysis.

Inconsistent System

An inconsistent system occurs when the set of equations has no solution. This typically occurs when the equations represent parallel lines or planes, meaning they never intersect. A clear signal of an inconsistent system is when you transform the augmented matrix and end up with a row where the variables cancel out, but the resulting constant does not equal zero, like what we found earlier: \[ \begin{array}{c} [0, 0, 0 | 4] \end{array} \] This row translates to the equation \(0 = 4\), which is a contradiction because no value of \(x\), \(y\), or \(z\) can satisfy this equation. Therefore, the system is inconsistent. This means there are no values that simultaneously satisfy all equations in the given system. When faced with an inconsistent system, it's essential to understand that it's not due to calculation errors but due to the nature of the equations themselves.