26 P Question: For the following, wri... [FREE SOLUTION]

Chapter 3: 26 P (page 137)

Question: For the following, write the solution in vector form.
${\begin{matrix} 2 x - 3 y + 5 z = 3 \\ x + 2 y - z = 5 \\ x - 5 y + 6 z = - 2 \\ 4 x + y + 3 z = 13 \end{matrix}$

Short Answer

Expert verified

The solution of the system of equations in the vector form is $r = (3,1,0) + (- 1,1,1) z$ .

Step by step solution

Definition of row reduction

Row reduction is just a systematic way of taking linear combinations of the given equations to produce a simpler but equivalent set of equations using elementary row operations.

Solution of the equations

The solution is to be found for the given system of equations.

${\begin{matrix} 2 x - 3 y + 5 z = 3 \\ x + 2 y - z = 5 \\ x - 5 y + 6 z = - 2 \\ 4 x + y + 3 z = 13 \end{matrix}$

Write the equations in matrix form.

$(\begin{matrix} 2 & - 3 & 5 & 3 \\ 1 & 2 & - 1 & 5 \\ 1 & - 5 & 6 & - 2 \\ 4 & 1 & 3 & 13 \end{matrix})$

Apply row reduction to find the solution.

$(\begin{matrix} 2 & - 3 & 5 & 3 \\ 1 & 2 & - 1 & 5 \\ 1 & - 5 & 6 & - 2 \\ 4 & 1 & 3 & 13 \end{matrix}) \overset{R_{1} 鈬� R_{2}}{鈫�} (\begin{matrix} 1 & 2 & - 1 & 5 \\ 1 & - 3 & 5 & 3 \\ 1 & - 5 & 6 & - 2 \\ 4 & 1 & 3 & 13 \end{matrix}) {鈫�}_{R_{4} 鈫� R_{4} - 4 R_{1}}^{R_{2} 鈫� R_{2} - 2 R_{1}, R_{3} 鈫� R_{3} - R_{1}} (\begin{matrix} 1 & 2 & - 1 & 5 \\ 0 & - 7 & 7 & - 7 \\ 0 & - 7 & 7 & - 7 \\ 0 & - 7 & 7 & - 7 \end{matrix}) \overset{R_{2} 鈫� \frac{R_{2}}{- 7}}{鈫�} (\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & - 1 & 1 \\ 0 & - 7 & 7 & - 7 \\ 0 & - 7 & 7 & - 7 \end{matrix}) {鈫�}_{R_{4} 鈫� R_{4} + 7 R_{2}}^{R_{3} 鈫� R_{3} + 7 R_{2}} (\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$

Thus, the solution is $x = 3 - z$ and $y = 1 + z$ .

In the vector form the solution is $r = (3 - z,1 + z, z)$ or $r = (3,1,0) + (- 1,1,1) z$ .

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Question: For the following, write the solution in vector form.
${\begin{matrix} 2 x - 3 y + 5 z = 3 \\ x + 2 y - z = 5 \\ x - 5 y + 6 z = - 2 \\ 4 x + y + 3 z = 13 \end{matrix}$

Short Answer

Step by step solution

Definition of row reduction

Solution of the equations

One App. One Place for Learning.

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91影视

Question: For the following, write the solution in vector form.{2x-3y+5z=3x+2y-z=5x-5y+6z=-24x+y+3z=13

Short Answer

Step by step solution

Definition of row reduction

Solution of the equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Engineering Physics

Fundamentals of Physics

Kinematics Physics

Conservation of Energy and Momentum

Nuclear Physics

Study anywhere. Anytime. Across all devices.

Question: For the following, write the solution in vector form.
${\begin{matrix} 2 x - 3 y + 5 z = 3 \\ x + 2 y - z = 5 \\ x - 5 y + 6 z = - 2 \\ 4 x + y + 3 z = 13 \end{matrix}$