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Chapter 10: Problem 31

Find the complete solution of the linear system, or show that it is inconsistent. \(\left\\{\begin{aligned} 2 x+3 y-z &=1 \\ x+2 y &=3 \\ x+3 y+z &=4 \end{aligned}\right.\)

Short Answer

Expert verified
The solution is \((x, y, z) = (7, -2, 3)\).

Step by step solution

01

Write the Augmented Matrix

First, convert the system of equations into an augmented matrix format. The system \( \begin{align*} 2x + 3y - z &= 1 \ x + 2y &= 3 \ x + 3y + z &= 4 \end{align*} \) becomes the augmented matrix: \[\begin{bmatrix}2 & 3 & -1 & | & 1 \1 & 2 & 0 & | & 3 \1 & 3 & 1 & | & 4\end{bmatrix}\]
02

Perform Row Operations to Simplify

Perform row operations to simplify the matrix. Start by eliminating the first element below the pivot in the first column:Subtract row 2 from row 1: \( R_1 - 2R_2 \rightarrow R_1 \)\[\begin{bmatrix}0 & 1 & -1 & | & -5 \1 & 2 & 0 & | & 3 \1 & 3 & 1 & | & 4\end{bmatrix}\]Subtract row 2 from row 3 to eliminate \( x \) from row 3:\( R_3 - R_2 \rightarrow R_3 \)\[\begin{bmatrix}0 & 1 & -1 & | & -5 \1 & 2 & 0 & | & 3 \0 & 1 & 1 & | & 1\end{bmatrix}\]
03

Continue Eliminating to Row Echelon Form

Continue simplifying the matrix to row-echelon form. Subtract row 1 from row 3 to eliminate \( y \) from row 3:\( R_3 - R_1 \rightarrow R_3 \)\[\begin{bmatrix}0 & 1 & -1 & | & -5 \1 & 2 & 0 & | & 3 \0 & 0 & 2 & | & 6\end{bmatrix}\]
04

Solve the Simplified Matrix

Now solve the matrix. Start with the last equation:\( 2z = 6 \rightarrow z = 3 \).Substitute \( z = 3 \) into the first row:\( y - 3 = -5 \rightarrow y = -2 \).Finally, substitute \( y = -2 \) back into the second original equation:\( x + 2(-2) = 3 \rightarrow x - 4 = 3 \rightarrow x = 7 \).
05

Verify the Solution

To ensure the solution \( (x, y, z) = (7, -2, 3) \) satisfies all original equations, substitute these values back:For the first equation: \( 2(7) + 3(-2) - 3 = 1 \), which holds true.For the second equation: \( 7 + 2(-2) = 3 \), which holds true.For the third equation: \( 7 + 3(-2) + 3 = 4 \), which holds true as well.All equations are satisfied, confirming the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Augmented Matrix
An augmented matrix is an efficient way to organize a system of linear equations to solve it using matrix methods. In our example, the given system of equations:
  • \( 2x + 3y - z = 1 \)
  • \( x + 2y = 3 \)
  • \( x + 3y + z = 4 \)
is represented as an augmented matrix:\[\begin{bmatrix}2 & 3 & -1 & | & 1 \1 & 2 & 0 & | & 3 \1 & 3 & 1 & | & 4\end{bmatrix}\]This matrix includes coefficients from the equations on the left-hand side and the constants on the right-hand side (after the vertical line). This format makes it easy to apply matrix operations while keeping track of the constants from the equations.
Row Operations
Row operations are techniques used to manipulate an augmented matrix into a simpler form. These operations include:
  • Swapping rows.
  • Multiplying a row by a non-zero scalar.
  • Adding or subtracting multiples of one row from another row.
In our exercise, we first performed several row operations to simplify the augmented matrix and make it easier to solve. For example, we subtracted the second row from the first and third rows to help eliminate variables systematically. Through these operations, we aimed to create zeros below the leading '1's in each column, which simplifies the matrix towards a row-echelon form for easier solving.
Row-Echelon Form
Row-echelon form is a simplified form of the matrix where each leading entry (also known as a pivot) is to the right of the pivot in the row above. The matrix is simplified further so that solving becomes straightforward. Our augmented matrix was transformed as follows:\[\begin{bmatrix}0 & 1 & -1 & | & -5 \1 & 2 & 0 & | & 3 \0 & 0 & 2 & | & 6\end{bmatrix}\]In this example, each leading coefficient is followed by zeros below it, making it easier to solve the system using back-substitution, which is the next step in the process. In row-echelon form, you can solve the equation from the bottom row upwards, substituting known values back into previous rows.
Solution Verification
Once a solution for a system of equations is found, solution verification ensures that the solution satisfies all original equations. For our calculated solution \( (x, y, z) = (7, -2, 3) \), each variable substitution should make each equation true. To verify:
  • Substitute into the first equation: \[ 2(7) + 3(-2) - 3 = 1 \]
  • Check the second equation:\[ 7 + 2(-2) = 3 \]
  • And the third equation:\[ 7 + 3(-2) + 3 = 4 \]
Each of these calculations confirms the correctness of our solution. Verification helps ensure there was no arithmetic mistake during earlier steps and confirms the solution's accuracy.

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Most popular questions from this chapter

Graph the solution set of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l} x^{2}+y^{2} \leq 4 \\ x^{2}-2 y>1 \end{array}\right.$$

Airplane Speed A man flies a small airplane from Fargo to Bismarck, North Dakota- a distance of \(180 \mathrm{mi}\). Because he is flying into a headwind, the trip takes him 2 h. On the way back, the wind is still blowing at the same speed, so the return trip takes only \(1 \mathrm{h} 12\) min. What is his speed in still air, and how fast is the wind blowing?

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Graph the solution set of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x^{2}+y^{2} & \leq 4 \\ x-y &>0 \end{aligned}\right.$$
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