A linear system is called:[br][list][*][b][i]consistent[/i][/b] if it has at least one solution. In particular it is [i][b]independent [/b][/i]if it has exactly one solution and [i][b]dependent [/b][/i]if it has infinitely many solutions.[/*][*][i][b]inconsistent[/b][/i] if it has no solutions.[/*][/list]
Given a system [math]Ax=b[/math] of [math]n[/math] linear equations for [math]n[/math] unknowns, if the determinant [math]D[/math] of the matrix [math]A[/math] is nonzero, the system has a unique solution, given by:[br][math]x_1=\frac{D_1}{D}[/math], [math]x_2=\frac{D_2}{D},\ldots x_n=\frac{D_n}{D}[/math] [br]where [math]D_1,D_2,\ldots D_n[/math] are the determinants of the matrices obtained by replacing the [math]i[/math]-th column of [math]A[/math] with the column vector [math]b[/math].
Write the following system in matrix form, then determine whether it can be solved using the Cramer's Rule.[br][math]\begin{cases} 3x + 5y =9 \\ 7x-2y =10 \end{cases}[/math]
Which value of parameter [math]k[/math] makes the following system independent?[br][math]\begin{cases} x - y -3z =8 \\ 3x+ky-z =4 \\2x+3y+4z =-4 \end{cases}[/math][br]
Solve the following system using the Cramer's Rule.[br][math]\begin{cases} -x + 5y=2 \\ 7x-2y=0 \end{cases}[/math]
If applicable, solve the system [math]Ax=b[/math] using Cramer's Rule, given:[br][math]A=\begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix}[/math] and [math]b=\begin{pmatrix} 6 \\ -2 \end{pmatrix}[/math]
Solve the following system using the Cramer's Rule.[br][math]\begin{cases} x + y+z=6 \\ x-2y-z=-6 \\ 3x+3y-z=6 \end{cases}[/math]
If applicable, solve the system [math]Ax=b[/math] using the Cramer's Rule, given:[br][math]A=\begin{pmatrix} 1 & 1 & -1 \\ 4 & -1 & -5 \\ 1 & -4 & -2 \end{pmatrix}[/math] and [math]b=\begin{pmatrix} 5 \\ 6 \\ -4 \end{pmatrix}[/math]