Chapter 11: Q. 24 (page 793)

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.
$\{\begin{array}{r} x - y + z = 0 \\ x - y - 5 z - 6 = 0 \\ 2 x - 2 y + z - 1 = 0 \end{array}$

Short Answer

Expert verified

The solution is $\{(x, y, z) | x = y + 1, z = - 1, y i s a n y r e a l n u m b e r\}$ .

Step by step solution

Step 1. Given information

The given system of equations is $\{\begin{array}{r} x - y + z = 0 \\ x - y - 5 z - 6 = 0 \\ 2 x - 2 y + z - 1 = 0 \end{array}$

Step 2. Solve system of equations using matrix

Write the augmented matrix that represents the system of equations

$[\begin{array}{rrrr} 1 & - 1 & 1 & 0 \\ 1 & - 1 & - 5 & 6 \\ 2 & - 2 & 1 & 1 \end{array}]$

Perform the operations $R_{2} = R_{2} - R_{1}$ ,

$[\begin{array}{cccc} 1 & - 1 & 1 & 0 \\ 1 - 1 & - 1 - (- 1) & - 5 - (1) & 6 - 0 \\ 2 & - 2 & 1 & 1 \end{array}] \to [\begin{array}{crrc} 1 & - 1 & 1 & 0 \\ 0 & 0 & - 6 & 6 \\ 2 & - 2 & 1 & 1 \end{array}]$

Now, perform the operations $R_{3} = R_{3} - 2 R_{1}$ ,

$[\begin{array}{cccc} 1 & - 1 & 1 & 0 \\ 0 & 0 & - 6 & 6 \\ 2 - 2 (1) & - 2 - 2 (- 1) & 1 - 2 (1) & 1 - 2 (0) \end{array}] \to [\begin{array}{rrrr} 1 & - 1 & 1 & 0 \\ 0 & 0 & - 6 & 6 \\ 0 & 0 & - 1 & 1 \end{array}]$

The system of equations corresponding to this matrix is

$\{x - y + z = 0 - 6 z = 6 - z = 1$

So,

$z = - 1 x - y + (- 1) = 0 x - y = 1 x = y + 1$

Therefore, the solution is $\{(x, y, z) | x = y + 1, z = - 1, y i s a n y r e a l n u m b e r\}$

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In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.
$\{\begin{array}{r} x - y + z = 0 \\ x - y - 5 z - 6 = 0 \\ 2 x - 2 y + z - 1 = 0 \end{array}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve system of equations using matrix

One App. One Place for Learning.

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In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.x−y+z=0x−y−5z−6=02x−2y+z−1=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve system of equations using matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.
$\{\begin{array}{r} x - y + z = 0 \\ x - y - 5 z - 6 = 0 \\ 2 x - 2 y + z - 1 = 0 \end{array}$