Systems of linear equations and inequalities

9000019910

Level: 
B
The augmented matrix of a system of three equations with three unknowns is row equivalent to the following matrix \(A'\). Find the solution of the system. \[ A' = \left(\begin{array}{ccc|c} -1 & -6 & 1 &-20\\ 0 & 5 & 4 & -12\\ 0 & 0 & 0 & -8 \end{array}\right) \]
no solution
\(\left [-\frac{172} {5} ;-\frac{12} {5} ;0\right ]\)
\([-12t;4t;-8t],\ t\in \mathbb{R}\)
\(\left [-12;4;-8\right ]\)

9000019904

Level: 
B
The coefficient matrix of a \(3\times 3\) linear system is \(A\) and the augmented matrix \(A'\). Find \(\mathop{\mathrm{rank}}(A)\) and \(\mathop{\mathrm{rank}}(A')\). \[ A = \begin{pmatrix} -1 & 3 & 2 \\ 0 & 4 & -5 \\ 0 & 0 & 2 \end{pmatrix} \qquad A' = \left(\begin{array}{ccc|c} -1 & 3 & 2 & 5 \\ 0 & 4 & -5 & 10\\ 0 & 0 & 2 & 0 \end{array}\right) \]
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)

9000019905

Level: 
B
Let \(A\) and \(A'\) be the coefficient matrix and the augmented matrix of the following linear system, respectively. Find the ranks of these matrices. \[ \begin{array}{cl} \phantom{ -} 3x + 5y +\phantom{ 2}z =\phantom{ -}10& \\ - 2x - 3y + 2z = -10& \\ \phantom{ - 2}x +\phantom{ 2}y - 5z =\phantom{ -}10& \end{array} \]
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)

9000019906

Level: 
B
Consider a linear system of four equations with four unknowns. The rank of the coefficient matrix \(A\) is \(\mathop{\mathrm{rank}}(A) = 3\). The rank of the augmented matrix \(A'\) is \(\mathop{\mathrm{rank}}(A') = 4\). Identify a true statement on this system.
The system does not have any solution.
The system has infinitely many solutions.
The system has a unique solution.
It is not possible to draw any conclusion from this information.

9000019907

Level: 
B
The augmented matrix of a system of three equations with three unknowns is row equivalent with the following matrix \(A'\). Find the solution of the system. \[ A' = \left(\begin{array}{ccc|c} 1 & 2 & 4 & 0\\ 0 & 2 & 7 & 7\\ 0 & 0 & 7 & 35 \end{array}\right) \]
\([8;-14;5]\)
\([-62;21;5]\)
\([8;14;-5]\)
\([-22;-21;5]\)

9000019908

Level: 
B
The augmented matrix of a system of three equations with three unknowns is row equivalent with the following matrix \(A'\). Find the solution of the system. \[ A' = \left(\begin{array}{ccc|c} -1 & 0 & 1 &-1\\ 0 & 7 & 2 & -1\\ 0 & 0 & 30 & 6 \end{array}\right) \]
\(\left [\frac{6} {5};-\frac{1} {5}; \frac{1} {5}\right ]_{}\)
\(\left [\frac{1} {5};-\frac{1} {5}; \frac{6} {5}\right ]\)
\(\left [\frac{1} {5};-\frac{6} {5};-\frac{1} {5}\right ]\)
\(\left [-\frac{6} {5}; \frac{1} {5}; \frac{1} {5}\right ]\)

9000007206

Level: 
A
Consider the linear system: \[ \begin{aligned}2x - 3y - 12& = 0,& \\\text{???}\quad & = 0. \\ \end{aligned} \] In the following list, identify the missing second equation if you know that the system does not have a solution.
\(- 6x + 9y - 9 = 0\)
\(2x + 3y - 6 = 0\)
\(- 4x + 6y + 24 = 0\)
\(x + 2y - 12 = 0\)