Chapter 11: Q. 54 (page 730)

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 2 x - 3 y - z = 0 \\ 3 x + 2 y + 2 z = 2 \\ x + 5 y + 3 z = 2 \end{cases}$

Short Answer

Expert verified

The solution of the system is $x = \frac{6}{13} - \frac{4}{13} z, y = \frac{4}{13} - \frac{7}{13} z$ where, z is any real number.

Step by step solution

Step 1. Given information.

The given system of equation are

$\{\begin{cases} 2 x - 3 y - z = 0 \\ 3 x + 2 y + 2 z = 2 \\ x + 5 y + 3 z = 2 \end{cases}$

Step 2. Calculation.

The augmented matrix of the system is:

$[\begin{matrix} \begin{matrix} 2 & - 3 & - 1 \\ 3 & 2 & 2 \\ 1 & 5 & 3 \end{matrix} & \begin{matrix} 0 \\ 2 \\ 2 \end{matrix} \end{matrix}]$

Perform the row operations $R_{2} = r_{2} - r_{1}$ :

$[\begin{matrix} \begin{matrix} 2 & - 3 & - 1 \\ 1 & 5 & 3 \\ 1 & 5 & 3 \end{matrix} & \begin{matrix} 0 \\ 2 \\ 2 \end{matrix} \end{matrix}]$

Perform the row operations $R_{2} = r_{2} - \frac{1}{2} r_{1}, R_{3} = r_{3} - r_{2}$ :

$[\begin{matrix} \begin{matrix} 2 & - 3 & - 1 \\ 0 & \frac{13}{2} & \frac{7}{2} \\ 0 & 0 & 0 \end{matrix} & \begin{matrix} 0 \\ 2 \\ 0 \end{matrix} \end{matrix}]$

Perform the row operations $R_{2} = 2 r_{2}$ :

$[\begin{matrix} \begin{matrix} 2 & - 3 & - 1 \\ 0 & 13 & 7 \\ 0 & 0 & 0 \end{matrix} & \begin{matrix} 0 \\ 4 \\ 0 \end{matrix} \end{matrix}]$

The above matrix is in reduced row echelon form. The corresponding system of equations is

$2 x - 3 y - z = 0 (1) 13 y + 7 z = 4 (2)$

Step 3. Solve the equation.

From equation (2):

$13 y + 7 z = 4 13 y = 4 - 7 z y = \frac{4}{13} - \frac{7}{13} z$

Substitute $y = \frac{4}{13} - \frac{7}{13} z$ into equation (1).

$2 x - 3 y - z = 0 2 x - 3 (\frac{4}{13} - \frac{7}{13} z) - z = 0 2 x - \frac{12}{13} + \frac{21}{13} z - z = 0 2 x + \frac{8}{13} z = \frac{12}{13} 2 x = \frac{12}{13} - \frac{8}{13} z x = \frac{6}{13} - \frac{4}{13} z$

Therefore, the solution of the system is $x = \frac{6}{13} - \frac{4}{13} z, y = \frac{4}{13} - \frac{7}{13} z$ , where, z is any real number.

Step 4. Conclusion.

The solution of the system is $x = \frac{6}{13} - \frac{4}{13} z, y = \frac{4}{13} - \frac{7}{13} z$ where, z is any real number.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 2 x - 3 y - z = 0 \\ 3 x + 2 y + 2 z = 2 \\ x + 5 y + 3 z = 2 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Solve the equation.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.2x-3y-z=03x+2y+2z=2x+5y+3z=2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Solve the equation.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 2 x - 3 y - z = 0 \\ 3 x + 2 y + 2 z = 2 \\ x + 5 y + 3 z = 2 \end{cases}$