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Chapter 1: Problem 3
Determine whether the equation is linear in the variables \(x\) and \(y\). $$\frac{3}{y}+\frac{2}{x}-1=0$$
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Solve the system of linear equations for \(x\) and \(y\). $$\begin{aligned} (\cos \theta) x+(\sin \theta) y &=1 \\ (-\sin \theta) x+(\cos \theta) y &=1 \end{aligned}$$
Determine conditions on \(a, b, c,\) and \(d\) such that the matrix $$\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$ will be row-equivalent to the given matrix. $$\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]$$
(a) determine the polynomial function whose graph passes through the given points, and (b) sketch the graph of the polynomial function, showing the given points. $$(2006,5),(2007,7),(2008,12)(z=x-2007)$$
Find the solution set of the system of linear equations represented by the augmented matrix. $$\left[\begin{array}{rrrr} 1 & 2 & 1 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$
Consider the matrix \(A=\left[\begin{array}{rrr}2 & -1 & 3 \\ -4 & 2 & k \\ 4 & -2 & 6\end{array}\right]\) (a) If \(A\) is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. (b) If \(A\) is the augmented matrix of a system of linear equations, find the value(s) of \(k\) such that the system is consistent. (c) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, determine the number of equations and the number of variables. (d) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, find the value(s) of \(k\) such that the system is consistent.
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