Systems of linear equations and inequalities

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

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?
\( 420\,\mathrm{CZK} \)
\( 350\,\mathrm{CZK} \)
\( 440\,\mathrm{CZK} \)
\( 550\,\mathrm{CZK} \)

2010011201

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given inequalities. Which inequality is it?
\( y \geq -\frac32x+\frac{7}2 \)
\( y \leq -\frac32x+\frac{7}2 \)
\( y > -\frac32x+\frac{7}2 \)
\( y < -\frac32x+\frac{7}2 \)

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

2010006503

Level: 
A
Consider the linear system: \[ \begin{aligned}6x - 3y - 42& = 0,& \\\text{???}\quad & = 0. \\ \end{aligned} \] In the following list, identify the missing second equation if you know that the system does not have a solution.
\(- 2x + y +12 = 0\)
\( 2x + y +21 = 0\)
\(3x -2y -12 = 0\)
\(12x -6 y -84 = 0\)

2010006502

Level: 
A
Identify which of the following systems of equations has infinitely many solutions.
\( \begin{aligned} \frac12x-3y&=12\\ -\frac{1}3x+2y&=-8 \end{aligned} \)
\( \begin{aligned} \frac13 x-2y&=12 \\ -\frac12 x+3y&=-16 \end{aligned} \)
\( \begin{aligned} \frac12 x+2y&=12 \\ -\frac13 x-3y&=-12 \end{aligned} \)
\( \begin{aligned} \frac12 x-y&=12 \\ -\frac23 x+4y&=-8\end{aligned} \)

2010006501

Level: 
A
In \( \mathbb{R}\times\mathbb{R} \), find the solution set of the equation: \[ 3y-\frac{x+y}2=1-\frac43x \]
\( \left\{ \left[-3y+\frac65;y\right],\ y\in\mathbb{R}\right \} \)
\( \left\{ \left[-3y+\frac65;x+\frac13\right],\ x\in\mathbb{R},y\in\mathbb{R}\right \} \)
\( \left\{ \left[\frac13 y+\frac65;y\right],\ y\in\mathbb{R}\right \} \)
\( \emptyset \)

2000006804

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given inequalities. Which of the inequalities is it?
\[\begin{aligned} y &\leq x \\y &\geq -x \end{aligned}\]
\[\begin{aligned} y &\leq - x \\y &\geq x \end{aligned}\]
\[\begin{aligned} y &\leq x \\y &\leq -x \end{aligned}\]
\[\begin{aligned} y &\geq x \\y &\geq -x \end{aligned}\]