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Chapter 1: Problem 2
Determine whether the equation is linear in the variables \(x\) and \(y\). $$3 x-4 x y=0$$
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Solve the homogeneous linear system corresponding to the coefficient matrix provided. $$\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$
Find the values of \(x, y,\) and \(\lambda\) that satisfy the system of equations. Such systems arise in certain problems of calculus, and \(\lambda\) is called the Lagrange multiplier. $$\begin{array}{rr} 2 y+2 \lambda+\quad 2= & 0 \\ 2 x \quad \lambda+\quad 1=0 \\ 2 x+y \quad-100=0 \end{array}$$
Determine the size of the matrix. $$\left[\begin{array}{rrrrr} 8 & 6 & 4 & 1 & 3 \\ 2 & 1 & -7 & 4 & 1 \\ 1 & 1 & -1 & 2 & 1 \\ 1 & -1 & 2 & 0 & 0 \end{array}\right]$$
Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\begin{aligned} x_{1}+x_{2}-5 x_{3} =3 \\ x_{1} -2 x_{3}=1 \\ 2 x_{1}-x_{2}-x_{3} =0 \end{aligned}$$
Is it possible for a system of linear equations with fewer equations than variables to have no solution? If so, give an example.
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