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.

$(\begin{matrix} 2 & - 3 & 5 & 3 \\ 1 & 2 & - 1 & 5 \\ 1 & - 5 & 6 & - 2 \\ 4 & 1 & 3 & 13 \end{matrix}) \overset{R_{1} \Leftrightarrow R_{2}}{\to} (\begin{matrix} 1 & 2 & - 1 & 5 \\ 1 & - 3 & 5 & 3 \\ 1 & - 5 & 6 & - 2 \\ 4 & 1 & 3 & 13 \end{matrix}) \to_{R_{4} \to R_{4} - 4 R_{1}}^{R_{2} \to R_{2} - 2 R_{1}, R_{3} \to R_{3} - R_{1}} (\begin{matrix} 1 & 2 & - 1 & 5 \\ 0 & - 7 & 7 & - 7 \\ 0 & - 7 & 7 & - 7 \\ 0 & - 7 & 7 & - 7 \end{matrix}) \overset{R_{2} \to \frac{R_{2}}{- 7}}{\to} (\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & - 1 & 1 \\ 0 & - 7 & 7 & - 7 \\ 0 & - 7 & 7 & - 7 \end{matrix}) \to_{R_{4} \to R_{4} + 7 R_{2}}^{R_{3} \to R_{3} + 7 R_{2}} (\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$

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

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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

Medical Physics

Energy Physics

Fluids

Geometrical and Physical Optics

Quantum Physics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

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