Systems of linear equations and inequalities

2010006702

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. \[ A' = \left(\begin{array}{ccc|c} 2 & 3 & 1 & 7\\ 0 & 3 & 4 & 0\\ 0 & 0 & 5 & 45 \end{array}\right) \]
\([17;-12;9]\)
\([12;10;-9]\)
\([-19;12;9]\)
\([7;0;45]\)

2010011203

Level: 
B
The March price of a T-shirt and shorts was \( 900\,\mathrm{CZK} \) together. In April there was on store price adjustment. The price of the shorts decreased by \( 20\% \) and the price of the T-shirt increased by \( 20\% \). So the April price of both together the shorts and the T-shirt was by \( 40\,\mathrm{CZK} \) lower. What was the April price of the T-shirt?
\( 420\,\mathrm{CZK} \)
\( 350\,\mathrm{CZK} \)
\( 440\,\mathrm{CZK} \)
\( 550\,\mathrm{CZK} \)

2010011204

Level: 
B
Kamil is able to mow a meadow in \( 12 \) hours. Zdeněk has a better lawn mower and he is able to mow the same meadow in \( 9 \) hours. They have agreed that Kamil starts to mow alone sooner and Zdeněk will join him later so that the total time of mowing is \( 8 \) hours. How long will they mow together?
\( 3 \) hours
\( 5 \) hours
\( 2 \) hours
\( 1 \) hour

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. \[ A' = \left(\begin{array}{ccc|c} -1 & 0 & 1 &-1\\ 0 & 7 & 2 & -1\\ 0 & 0 & 30 & 6 \end{array}\right) \]
\(\left [\frac{6} {5};-\frac{1} {5}; \frac{1} {5}\right ]_{}\)
\(\left [\frac{1} {5};-\frac{1} {5}; \frac{6} {5}\right ]\)
\(\left [\frac{1} {5};-\frac{6} {5};-\frac{1} {5}\right ]\)
\(\left [-\frac{6} {5}; \frac{1} {5}; \frac{1} {5}\right ]\)

9000019909

Level: 
B
The augmented matrix of a system of three equations with three unknowns is the following matrix \(M'\). Identify the matrix which is row equivalent to \(M'\). \[ M' = \left(\begin{array}{ccc|c} 1 & 2 & 4 & 14\\ -1 & 0 & 3 & 7\\ 3 & 1 & -2 & 42 \end{array}\right) \]
\(\left(\begin{array}{ccc|c} 1 & 2 & 4 & 14\\ 0 & 2 & 7 & 21\\ 0 & 0 & 7 & 105 \end{array}\right)\)
\(\left(\begin{array}{ccc|c} 1 & 2 & 4 & 14\\ 0 & 2 & 7 & 21\\ 0 & 0 & -8 & 70 \end{array}\right)\)
\(\left(\begin{array}{ccc|c} 1 & 2 & 4 & 14\\ 0 & 2 & 7 & 21\\ 0 & 0 & -29 & -147 \end{array}\right)\)
\(\left(\begin{array}{ccc|c} 1 & 2 & 4 & 14\\ 0 & 2 & 1 & 7\\ 0 & 0 & -23 & 35 \end{array}\right)\)