Chapter 11: Q. 40 (page 740)

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.
$\{\begin{cases} x + 4 y - 3 z = 0 \\ 3 x - y + 3 z = 0 \\ x + y + 6 z = 0 \end{cases}$

Short Answer

Expert verified

The solution of the system of equations $\{\begin{cases} x + 4 y - 3 z = 0 \\ 3 x - y + 3 z = 0 \\ x + y + 6 z = 0 \end{cases}$ is $(0, 0, 0)$

Step by step solution

Step 1. Given information

The given system of equations is

$\{\begin{cases} x + 4 y - 3 z = 0 \\ 3 x - y + 3 z = 0 \\ x + y + 6 z = 0 \end{cases}$

Step 2. Determinant

The determinant D of the coefficients of the variables

$D = |\begin{matrix} 1 & 4 & - 3 \\ 3 & - 1 & 3 \\ 1 & 1 & 6 \end{matrix}| D = {(- 1)}^{1 + 1} (1) |\begin{matrix} - 1 & 3 \\ 1 & 6 \end{matrix}| + {(- 1)}^{1 + 2} (4) |\begin{matrix} 3 & 3 \\ 1 & 6 \end{matrix}| + {(- 1)}^{1 + 3} (- 3) |\begin{matrix} 3 & - 1 \\ 1 & 1 \end{matrix}| D = 1 (- 6 - 3) - 4 (18 - 3) - 3 (3 + 1) D = - 9 - 60 - 12 D = - 81$

$D \neq 0,$ so Cramer's rule can be used to determine the solution of the system of equations.

so that $x = \frac{D_{x}}{D}, y = \frac{D_{y}}{D}, z = \frac{D_{z}}{D}$

Step 3. Solution of system

All constants of the system of the equation are zeros

$D_{x} = 0 D_{y} = 0 D_{z} = 0$

Determinants $D \neq 0$ and all constants of the system of the equation are zeros so the system of equations has a trivial solution

$x = 0 y = 0 z = 0$

The solution of the system of equations is $(0, 0, 0)$

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In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.
$\{\begin{cases} x + 4 y - 3 z = 0 \\ 3 x - y + 3 z = 0 \\ x + y + 6 z = 0 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determinant

Step 3. Solution of system

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In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.x+4y-3z=03x-y+3z=0x+y+6z=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determinant

Step 3. Solution of system

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Discrete Mathematics

Probability and Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.
$\{\begin{cases} x + 4 y - 3 z = 0 \\ 3 x - y + 3 z = 0 \\ x + y + 6 z = 0 \end{cases}$