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Chapter 2: Problem 9

Solve the following system of simultaneous equations for \(x\). \(\begin{aligned}-2.0 x_{1}+5.0 x_{2}+1.0 x_{3}+3.0 x_{4}+4.0 x_{5}-1.0 x_{6}=0.0 \\ 2.0 x_{1}-1.0 x_{2}-5.0 x_{3}-2.0 x_{4}+6.0 x_{5}+4.0 x_{6}=& 1.0 \\\\-1.0 x_{1}+6.0 x_{2}-4.0 x_{3}-5.0 x_{4}+3.0 x_{5}-1.0 x_{6}=&-6.0 \\\ 4.0 x_{1}+3.0 x_{2}-6.0 x_{3}-5.0 x_{4}-2.0 x_{5}-2.0 x_{6}=10.0 \\\\-3.0 x_{1}+6.0 x_{2}+4.0 x_{3}+2.0 x_{4}-6.0 x_{5}+4.0 x_{6}=&-6.0 \\ 2.0 x_{1}+4.0 x_{2}+4.0 x_{3}+4.0 x_{4}+5.0 x_{5}-4.0 x_{6}=&-2.0 \end{aligned}\)

Short Answer

Expert verified
The solution to the given system of simultaneous equations is: \(x_{1} = -1.0\), \(x_{2} = -1.0\), \(x_{3} = 1.0\), \(x_{4} = 0.0\), \(x_{5} = -1.0\), and \(x_{6} = -1.0\).

Step by step solution

01

Write the augmented matrix

Before we start the Gaussian Elimination, we need to write the augmented matrix of the system of linear equations. The augmented matrix will have 7 columns, where the first 6 come from the coefficients of the variables and the 7th comes from the constant terms of the equations: $\left[ \begin{array}{c c c c c c | c} -2.0 & 5.0 & 1.0 & 3.0 & 4.0 & -1.0 & 0.0 \\ 2.0 & -1.0 & -5.0 & -2.0 & 6.0 & 4.0 & 1.0 \\ -1.0 & 6.0 & -4.0 & -5.0 & 3.0 & -1.0 & -6.0 \\ 4.0 & 3.0 & -6.0 & -5.0 & -2.0 & -2.0 & 10.0 \\ -3.0 & 6.0 & 4.0 & 2.0 & -6.0 & 4.0 & -6.0 \\ 2.0 & 4.0 & 4.0 & 4.0 & 5.0 & -4.0 & -2.0 \\ \end{array} \right]$
02

Transform the matrix into row-echelon form

We will perform row operations to reduce the augmented matrix into row-echelon form. These row operations include: - Swapping any two rows - Multiplying a row by a nonzero constant - Adding or subtracting a multiple of a row to/from another row This step is tedious and prone to calculation errors, but it is necessary to solve the system. After performing these row operations, you should achieve a matrix in row-echelon form. Note that the row-echelon form might not be unique.
03

Perform back-substitution

After obtaining the row-echelon form, we will perform back-substitution to find the values of the variables. Start from the last equation in row-echelon form and work your way back up, substituting values for variables when possible. Be sure to validate the values by checking if they satisfy the original system of equations. At the end of this process, you will have found the values of all the unknowns (if there is a unique solution), or you will have discovered that the system has an infinite number of solutions or no solution at all. In this case, please consult a textbook or a more advanced mathematical tool for guidance on the methodology for these cases.

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Most popular questions from this chapter

The distance between two points \((x 1, y 1)\) and \((x 2, y 2)\) on a Cartesian coordinate plane is given by the equation $$ d=\sqrt{\left(x l-x^{2}\right)^{2}+(y l-y 2)^{2}} $$ Write a program to calculate the distance between any two points \((x 1, y 1)\) and \((x 2, y 2)\) specified by the user. Use good programming practices in your program. Use the program to calculate the distance between the points \((2,3)\) and \((8,-5)\).

Hyperbolic cosine. The hyperbolic cosine function is defined by the equation $$ \cosh x=\frac{e^{x}+e^{-x}}{2} $$ Write a program to calculate the hyperbolic cosine of a user-supplied value \(x\). Use the program to calculate the hyperbolic cosine of \(3.0\). Compare the answer that your program produces to the answer produced by the MATLAB intrinsic function cosh \((x)\). Also, use MATLAB to plot the function \(\cosh (x)\). What is the smallest value that this function can have? At what value of \(x\) does it occur?

Answer the following questions for the array shown below. $$ \text { array } 1=\left[\begin{array}{rrrrr} 1.1 & 0.0 & 2.1 & -3.5 & 6.0 \\ 0.0 & 1.1 & -6.6 & 2.8 & 3.4 \\ 2.1 & 0.1 & 0.3 & -0.4 & 1.3 \\ -1.4 & 5.1 & 0.0 & 1.1 & 0.0 \end{array}\right] $$ a. What is the size of array2? b. What is the value of array \(1(4,2)\) ? c. What is the size and value of array \(1(:, 2: 2)\) ? d. What is the size and value of array \(1([13]\), end) ?

Radio Receiver. The voltage across the resistive load in Figure \(2.13\) varies as a function of frequency according to Equation (2.18). $$ V_{R}=\frac{R}{\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}} V_{\theta} $$ where \(\omega=2 \pi f\) and \(f\) is the frequency in hertz. Assume that \(L=0.1 \mathrm{mH}\), \(C=0.25 \mathrm{nF}, R=50 \Omega\), and \(V_{O}=10 \mathrm{mV}\). a. Plot the voltage on the resistive load as a function of frequency. At what frequency does the voltage on the resistive load peak? What is the voltage on the load at this frequency? This frequency is called the resonant frequency \(f_{0}\) of the circuit. b. If the frequency is changed to \(10 \%\) greater than the resonant frequency, what is the voltage on the load? How selective is this radio receiver? c. At what frequencies will the voltage on the load drop to half of the voltage at the resonant frequency?

Decibels. Engineers often measure the ratio of two power measurements in decibels, or \(\mathrm{dB}\). The equation for the ratio of two power measurements in decibels is $$ \mathrm{dB}=10 \log _{10} \frac{P_{2}}{P_{1}} $$ where \(P_{2}\) is the power level being measured, and \(P_{1}\) is some reference power level. a. Assume that the reference power level \(P_{1}\) is I milliwatt, and write a program that accepts an input power \(P_{2}\) and converts it into \(\mathrm{dB}\) with respect to the \(1 \mathrm{~mW}\) reference level. (Engineers have a special unit for dB power levels with respect to a \(1 \mathrm{~mW}\) reference: \(\mathrm{dBm}\).) Use good programming practices in your program. b. Write a program that creates a plot of power in watts versus power in \(\mathrm{dBm}\) with respect to a \(1 \mathrm{~mW}\) reference level. Create both a linear \(x y\) plot and a log-linear \(x y\) plot.
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