Problem 49 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{rr} 5 x-2 y+z= & 5 \\ x+y-2 z= & -2 \\ 4 x-3 y+3 z= & 7 \end{array} $$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Write the Augmented Matrix

Represent the given system of equations as an augmented matrix. The system is: \( \begin{aligned} 5x - 2y + z &= 5 \ x + y - 2z &= -2 \ 4x - 3y + 3z &= 7 \end{aligned} \). The augmented matrix is: \[ \left[ \begin{array}{ccc|c} 5 & -2 & 1 & 5 \1 & 1 & -2 & -2 \4 & -3 & 3 & 7 \end{array} \right] \].

Step 2: Perform Row Operations to Triangular Form

Use row operations to simplify the matrix. First, eliminate the first column's second and third rows:Subtract \( \frac{1}{5} imes \text{Row 1} \) from \( \text{Row 2} \):\[ \left[ \begin{array}{ccc|c} 5 & -2 & 1 & 5 \0 & \frac{7}{5} & -\frac{11}{5} & -\frac{19}{5} \4 & -3 & 3 & 7 \end{array} \right] \]. Subtract \( \frac{4}{5} imes \text{Row 1} \) from \( \text{Row 3} \):\[ \left[ \begin{array}{ccc|c} 5 & -2 & 1 & 5 \0 & \frac{7}{5} & -\frac{11}{5} & -\frac{19}{5} \0 & -\frac{7}{5} & \frac{11}{5} & 3 \end{array} \right] \].

Step 3: Simplify Lower Rows Further

Add Row 2 to Row 3 to eliminate the second element of Row 3:\[ \left[ \begin{array}{ccc|c} 5 & -2 & 1 & 5 \0 & \frac{7}{5} & -\frac{11}{5} & -\frac{19}{5} \0 & 0 & 0 & -\frac{4}{5} \end{array} \right] \].

Step 4: Interpret the Result and Conclusion

The resulting augmented matrix shows that the third row corresponds to the equation \( 0x + 0y + 0z = -\frac{4}{5} \), which is a contradiction.This means there is no solution to the system.

Key Concepts

Augmented Matrix

To tackle a system of linear equations effectively, we often start by creating an augmented matrix. This matrix is a simplified representation of the equations, capturing all the necessary coefficients and constants. In our exercise, the system of equations is:

• 5x - 2y + z = 5
• x + y - 2z = -2
• 4x - 3y + 3z = 7

To convert these into an augmented matrix, we extract the coefficients of the variables and align them into matrix form, with an extra column on the right for the constants. This gives us: \[\left[ \begin{array}{ccc|c} 5 & -2 & 1 & 5 \ 1 & 1 & -2 & -2 \ 4 & -3 & 3 & 7 \end{array} \right]\]Here:

• The first three columns represent the coefficients of x, y, and z respectively.
• The fourth column shows the constants from the equations.

Creating an augmented matrix simplifies solving the system by focusing on numerical manipulation rather than symbolic ones.

Row Operations

Row operations are essential tools when working with augmented matrices. They help transform the matrix into a form that makes the solution clear or indicates that no solutions exist. There are three main types of row operations:

• Row swapping - exchanging two rows.
• Scaling a row by a non-zero factor - multiplying or dividing a whole row by a constant.
• Row addition/subtraction - adding or subtracting a multiple of one row to or from another row.

In the given solution, row operations were employed to simplify the matrix:1. Subtracting \(\frac{1}{5} \times \text{Row 1}\) from \(\text{Row 2}\) and \(\frac{4}{5} \times \text{Row 1}\) from \(\text{Row 3}\).2. Further transforming to eliminate elements in rows, making calculations more manageable.The goal is often to achieve a triangular form, where most matrix entries below the main diagonal are zeroes. This format makes it easier to identify solutions or contradictions in the system.

No Solution in Linear Systems

Sometimes, when working with linear systems, we may encounter a situation where no solution is possible. This is indicated by a row that implies a contradiction, such as an equation like \( 0x + 0y + 0z = b \) with \(b eq 0\).In our example, the row operations lead us to:\[\left[ \begin{array}{ccc|c} 5 & -2 & 1 & 5 \ 0 & \frac{7}{5} & -\frac{11}{5} & -\frac{19}{5} \ 0 & 0 & 0 & -\frac{4}{5} \end{array} \right]\]The third row implies \(0x + 0y + 0z = -\frac{4}{5}\), which is a clear contradiction (how can 0 equal a non-zero value?). This means the system of equations has no solutions because the equations represent parallel planes that do not intersect. Recognizing such scenarios saves time and helps avoid chasing non-existent solutions.