Problem 18 - Elementary and Intermediate Algebra

Question

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} 4 x+y+z &=17 ,\\\ x-3 y+2 z &=-8, \\ 5 x-2 y+3 z &=5 \end{aligned}$$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Write the system of equations

The given system of equations is: 1) \(4x + y + z = 17\)2) \(x - 3y + 2z = -8\)3) \(5x - 2y + 3z = 5\)

Step 2: Express the equations in matrix form

Write the augmented matrix for the system of equations: \(\begin{bmatrix} 4 & 1 & 1 &|& 17 \ 1 & -3 & 2 &|& -8 \ 5 & -2 & 3 &|& 5 \end{bmatrix}\)

Step 3: Apply row operations to achieve row echelon form

Perform row operations to simplify the system. For simplicity, here are the necessary row operations directly applied:1. Swap \(R1\) and \(R2\) to make the leading coefficient in \(R1\) equal to 1.2. Replace \(R2\) with \(R2 - 4R1\), and \(R3\) with \(R3 - 5R1\): \(\begin{bmatrix} 1 & -3 & 2 &|& -8 \ 0 & 13 & -7 &|& 49 \ 0 & 13 & -7 &|& 45 \end{bmatrix}\)3. Replace \(R3\) with \(R3 - R2\): \(\begin{bmatrix} 1 & -3 & 2 &|& -8 \ 0 & 13 & -7 &|& 49 \ 0 & 0 & 0 &|& -4 \end{bmatrix}\)

Step 4: Analyze the resulting matrix

The last row is \(0 = -4\), which is a contradiction. Therefore, the system of equations has no solution.

Key Concepts

augmented matrix

When solving systems of equations, it is useful to represent the system using an augmented matrix. An augmented matrix includes both the coefficients and the constants from the equations.
To create an augmented matrix for the given system: \(4x + y + z = 17\), \(x - 3y + 2z = -8\), and \(5x - 2y + 3z = 5\), align the coefficients and constants in rows: \[ \begin{bmatrix} 4 & 1 & 1 &|& 17 \ 1 & -3 & 2 &|& -8 \ 5 & -2 & 3 &|& 5 \end{bmatrix} \]. Each row represents one equation, and each column before the vertical line represents a variable's coefficient. The rightmost column contains the constants. This matrix serves as a compact, visual way to work with the system.

row operations

Row operations are the steps we use to simplify the augmented matrix. These operations help in transforming the matrix to a simpler form, ideally the row echelon form.
There are three types of row operations:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting the multiple of one row to another row

In our step-by-step solution:
The first operation swaps \(R1\) and \(R2\) to make our leading coefficient in the first row (now new \(R1\)) equal to 1: \[ \begin{bmatrix} 1 & -3 & 2 &|& -8 \ 4 & 1 & 1 &|& 17 \ 5 & -2 & 3 &|& 5 \end{bmatrix} \]
The second operation simplifies the second and third rows using the first: \[ \begin{bmatrix} 1 & -3 & 2 &|& -8 \ 0 & 13 & -7 &|& 49 \ 0 & 13 & -7 &|& 45 \end{bmatrix} \].
Note how, by using row operations, we made the matrix easier to work with.

row echelon form

Row echelon form (REF) is an important matrix form that simplifies solving linear systems. In REF:

• The leading coefficient (first non-zero number from left) in each row is always 1 (known as a leading 1).
• Each leading 1 is to the right of the leading 1 in the row above it.
• Any rows containing all zeroes are at the bottom.

We achieved a near row echelon form in our solution by applying row operations: \[ \begin{bmatrix} 1 & -3 & 2 &|& -8 \ 0 & 13 & -7 &|& 49 \ 0 & 0 & 0 &|& -4 \end{bmatrix} \].
Notice that the last row contains only zeroes on the left but yet has a constant \(-4\) on the right. This is a pivotal point for detecting whether a solution exists.

no solution

Sometimes, after transforming a system of equations into an augmented matrix and applying row operations, we can find there is no solution. In our example, the last row of the matrix became: \[ \begin{bmatrix} 0 & 0 & 0 &|& -4 \end{bmatrix} \].
This row translates to \0 = -4\, a contradiction since zero cannot equal a non-zero number. This inconsistency indicates that no set of values for \(x, y,\) and \(z\) can satisfy all three equations simultaneously. Thus, the system of equations has no solution, also known as being inconsistent.
Recognizing such contradictions quickly helps to conclude that a solution is impossible, saving further unnecessary computations.