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Chapter 11: Problem 1

Solve: $$ \begin{array}{r} x+3 y+5 z=0 \\ 7 x-y+z=0 \\ 2 x+3 y-8 z=0 \end{array} $$

Short Answer

Expert verified
x = 0, y = 0, z = 0.

Step by step solution

01

- Write the System of Equations

Given the system of equations:1) \( x + 3y + 5z = 0 \)2) \( 7x - y + z = 0 \)3) \( 2x + 3y - 8z = 0 \)
02

- Eliminate One Variable

Multiply the second equation by 3 to set up elimination of the variable y:\( 3 \times (7x - y + z) = 3 \times 0 \)This gives: \( 21x - 3y + 3z = 0 \)
03

- Add/Subtract Equations to Eliminate y

Add equation 1 and the result from Step 2 to eliminate y:\( (x + 3y + 5z) + (21x - 3y + 3z) = 0 + 0 \)This simplifies to: \( 22x + 8z = 0 \)
04

- Solve for One Variable

From Step 3, solve for x:\( 22x + 8z = 0 \)Rearrange: \( x = -\frac{8}{22}z = -\frac{4}{11}z \)
05

- Substitute x Back Into Another Equation

Substitute \( x = -\frac{4}{11}z \) into the second original equation:\( 7x - y + z = 0\)\( 7(-\frac{4}{11}z) - y + z = 0 \)This simplifies to: \( -\frac{28}{11}z - y + z = 0 \)Combine like terms: \( -\frac{17}{11}z - y = 0 \)
06

- Solve for Another Variable

From Step 5, solve for y:\( y = -\frac{17}{11}z \)
07

- Verify the Solution

Use the third original equation to verify z:\( 2x + 3y - 8z = 0 \)Substitute \( x = -\frac{4}{11}z \) and \( y = -\frac{17}{11}z \):\( 2(-\frac{4}{11}z) + 3(-\frac{17}{11}z) - 8z = 0 \)This simplifies to: \( -\frac{8}{11}z -\frac{51}{11}z - 8z = 0 \)Combine like terms and simplify: \( -\frac{8}{11}z - \frac{51}{11}z - \frac{88}{11}z = 0 \)This becomes: \( -\frac{147}{11}z = 0 \)Thus, \( z = 0 \)
08

- Find Final Values for All Variables

Since \( z = 0 \), substitute back to find x and y:\( x = -\frac{4}{11}(0) = 0 \)\( y = -\frac{17}{11}(0) = 0 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Linear Equations
A **system of linear equations** consists of two or more equations involving the same set of variables. Systems can appear in various forms, such as
  • Two equations with two variables
  • Three equations with three variables
The goal when solving a system is to find values for each variable that satisfy all the equations simultaneously. Consider the given system:
\(\begin{array}{r} \ x+3 y+5 z=0 \ 7 x-y+z=0 \ 2 x+3 y-8 z=0 \ \ \ \end{array}\)
These equations might represent different planes in three-dimensional space. The solution of the system is the point where all three planes intersect.
Gaussian Elimination
The **Gaussian elimination** method is a systematic approach for solving systems of linear equations. It involves transforming the system's augmented matrix into a row-echelon form using three types of row operations:
  • Swapping two rows
  • Multiplying a row by a nonzero scalar
  • Adding or subtracting a multiple of one row to another row
For the provided system:
  • Step 1: Write the system of equations.
  • Step 2: Multiply the second equation by 3 to eliminate y: \( 21x - 3y + 3z = 0 \).
  • Step 3: Add equations to eliminate y: \( 22x + 8z = 0 \).
  • Step 4: Solve for one variable: \( x = -\frac{4}{11}z \).
  • Step 5: Substitute back into another equation to find another variable.
Through these steps, Gaussian elimination simplifies the arithmetic, making it easier to solve the system.
Variable Elimination
**Variable elimination**, part of Gaussian elimination, focuses on systematically removing variables to simplify the system. For instance:
  • In Step 2, we multiply one equation so that adding it to another will cancel out y.
  • In Step 3, adding equations after multiplication cancels out y, giving us an equation with only x and z.
Next, substituting the solved variable back into the remaining equations helps eliminate variables step-by-step until all variables are found. Finally, always verify your results by plugging the variables back into the original equations.

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Most popular questions from this chapter

By applying Kirchhoff's law and Ohm's law to a circuit, we obtain the following equations: $$ \begin{aligned} 3 i_{1}-5 i_{2}+3 i_{3} &=7.5 \\ 2 i_{1}+i_{2}-7 i_{3} &=-17.5 \\ -10 i_{1}+4 i_{2}+5 i_{3} &=16 \end{aligned} $$ where \(i_{1}, i_{2}\) and \(i_{3}\) represent currents. By using the inverse matrix method, find the values of \(i_{1}, i_{2}\) and \(i_{3}\).

Find the values of \(t\) so that $$ \operatorname{det}\left(\begin{array}{rrr} 1 & 0 & 3 \\ 5 & t & -7 \\ 3 & 9 & t-1 \end{array}\right)=0 $$

The natural period, \(T\), of vibrations of a building is given by $$ T=\frac{2 \pi}{\sqrt{-\lambda}} $$ where \(\lambda\) is the eigenvalue of a given matrix A. Determine \(T\) for $$ A=\left(\begin{array}{cc} -15 & 5 \\ 10 & -10 \end{array}\right) $$

Evaluate i detA ii \(\operatorname{det} \mathbf{B}\) iii \(\operatorname{det} \mathbf{A} \times \operatorname{det} \mathbf{B}\) iv \(\operatorname{det}(\mathbf{A B})\) for the following: a \(\mathbf{A}=\left(\begin{array}{ll}1 & 3 \\ 5 & 6\end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr}3 & 7 \\ -1 & 5\end{array}\right)\) b \(\mathrm{A}=\left(\begin{array}{cc}-1 & 10 \\ 170 & 1.5\end{array}\right), \mathbf{B}=\left(\begin{array}{rr}-30 & -9.3 \\ 61 & -1.9\end{array}\right)\) c \(\mathrm{A}=\left(\begin{array}{cc}5 & -3 \\ 2.2 & -5.6\end{array}\right), \mathbf{B}=\left(\begin{array}{rr}-7.1 & -2.1 \\ -3.5 & -12.2\end{array}\right)\) What do you notice about your results to parts iii and iv.

Find \(\mathrm{A}^{\mathrm{T}}\) (A transposed) for the following: \(\mathbf{a} \mathbf{A}=\left(\begin{array}{rr}-2 & 3 \\ 1 & 5\end{array}\right) \quad\) b \(\mathbf{A}=\left(\begin{array}{rc}1 / 3 & -2 / 5 \\\ -3 / 7 & \pi\end{array}\right)\) \(\mathbf{c} \mathrm{A}=\left(\begin{array}{rrr}7 & 3 & 4 \\ 2 & 6 & 1 \\ -3 & -3 & 1\end{array}\right)\) \(\mathbf{d} \mathrm{A}=\left(\begin{array}{rr}1.17 & 1.36 \\ 9.39 & -1.45 \\\ 2.11 & 5.20\end{array}\right)\) e \(A=\left(\begin{array}{rrrr}-7 & 2 & 1 & 5 \\ 3 & 6 & -4 & 7 \\ 8 & 3 & -3 & 5 \\ -4 & 6 & 7 & 0\end{array}\right)\) f \(A=\left(\begin{array}{rrr}1 & -3 & 7 \\ -9 & 4 & 6 \\ -4 & 2 & 8 \\ 9 & 19 & 11\end{array}\right)\)
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