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Chapter 8: Problem 106
Determine whether the statement is true or false. Justify your answer. If a system of three linear equations is inconsistent, then its graph has no points common to all three equations.
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Consider the circuit in the figure. The currents \(I_{1}, I_{2},\) and \(I_{3},\) in amperes, are given by the solution of the system of linear equations \(\left\\{\begin{aligned} 2 I_{1} &+4 I_{3}=E_{1} \\ I_{2}+4 I_{3} &=E_{2} \\\ I_{1}+I_{2}-I_{3} &=0 \end{aligned}\right.\) where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. \(E_{1}=10\) volts, \(E_{2}=10\) volts
Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{llll} 1 & 8 & \vdots & 12 \\ 0 & 1 & \vdots & 3 \end{array}\right]$$
Solve for \(x\) $$\left|\begin{array}{cc} x+1 & 2 \\ -1 & x \end{array}\right|=4$$
Solve for \(x\) $$\left|\begin{array}{rrr} 1 & 2 & x \\ -1 & 3 & 2 \\ 3 & -2 & 1 \end{array}\right|=0$$
Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{rrrrr} 1 & -1 & 4 & \vdots & 0 \\ 0 & 1 & -1 & \vdots & 2 \\ 0 & 0 & 1 & \vdots & -2 \end{array}\right]$$
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