Systems of linear equations and inequalities

1003034503

Level: 
B
Students registered for sports camps. For the biking camp registered by \( 18 \) students more than for the boating camp. After some time one of the students switched his registration from the boating camp to the biking camp. Now, there is two times more bikers than boaters. How many students registered originally for the boating camp?
\( 21 \)
\( 39 \)
\( 20 \)
\( 15 \)

1003034502

Level: 
C
Petr would like to buy a new smartphone. If he starts a temporary job at the electro shop he gets payed \( 120\,\mathrm{CZK} \) per hour plus he gets \( 20\% \) discount on the smartphone bough in the shop. He calculated that for \( 24 \) hours of work he would not earn even half of the phone price. Another employer pays \( 150\,\mathrm{CZK} \) per hour. If Petr gets a temporary job with the other employer he is not eligible for the discount in the electro shop anymore, however he can buy the smartphone from e-shop for the price by \( 600\,\mathrm{CZK} \) lower than in the electro shop and for \( 20 \) hours of work he earns more than one third of the electro shop smartphone price. Determine as closely as possible the smartphone price in the electronics shop.
greater than \( 7\,200\,\mathrm{CZK} \) and less than \( 9\,600\,\mathrm{CZK} \)
greater than \( 7\,200\,\mathrm{CZK} \) and less than \( 10\,800\,\mathrm{CZK} \)
greater than \( 4\,800\,\mathrm{CZK} \) and less than \( 9\,600\,\mathrm{CZK} \)
greater than \( 4\,800\,\mathrm{CZK} \) and less than \( 10\,800\,\mathrm{CZK} \)

1003034501

Level: 
C
Two different aquarium fish shops have special price on Congo Tetra. The price is \( 42\,\mathrm{CZK} \) for one fish. Further, the discount of \( 50\,\mathrm{CZK} \) from a purchase over \( 300\,\mathrm{CZK} \) is offered in shop A. In shop B a customer is given \( 5\% \) discount of the price of any purchase. How many of Congo Tetra must one buy to make the total price in shop A lower than the total price in shop B?
greater than \( 7 \) and less than \( 24 \)
less than \( 24 \)
greater than \( 23 \)
less than \( 7 \)

1003083003

Level: 
A
Find the solution set of the following system of equations. \[ \begin{aligned}\frac23 x-\frac12y&=1 \\ -2x+\frac32y&=-3 \end{aligned} \]
\( \left\{\left[x; \frac{4x-6}3\right]\colon x\in\mathbb{R}\right\} \)
\( \left\{\left[x; y\right]\colon x\in\mathbb{R}\text{, } y\in\mathbb{R}\right\} \)
\( \emptyset \)
\( \left\{[0; -2]\right\} \)

1003083002

Level: 
A
Identify which of the sets is not the solution set of the following system of equations. \[ \begin{aligned} \frac12 x-y&=3 \\ \frac x3 - \frac23 y &=2 \end{aligned} \]
\( \left\{\left[6+2y;\frac{x-6}2\right]\colon x\in\mathbb{R}\text{, }y\in\mathbb{R}\right\} \)
\( \left\{\left[x; \frac{x-6}2\right]\colon x\in\mathbb{R}\right\} \)
\( \left\{\left[6+2y;y\right]\colon y\in\mathbb{R}\right\} \)
\( \left\{\left[2t;t-3\right]\colon t\in\mathbb{R}\right\} \)

1003083001

Level: 
A
Identify which of the following systems of equations has infinitely many solutions.
\( \begin{aligned} \frac13x-4y&=2\\ -\frac{x}4+3y&=-\frac32 \end{aligned} \)
\( \begin{aligned} \frac13 x-4y&=2 \\ -x+12y&=6 \end{aligned} \)
\( \begin{aligned} \frac13 x-4y&=2 \\ \frac x4-6y&=6 \end{aligned} \)
\( \begin{aligned} \frac13 x-4y&=2 \\ \frac x3-4y&=0 \end{aligned} \)

1003060504

Level: 
B
Four systems of equations are given. How many of the given systems have infinitely many solutions? \[ \begin{array}{c|c} \text{\( \begin{aligned} 4x-6y+10z&=8 \\ -2x+3y-5z&=4 \\ x+y+z&=1 \end{aligned}\)}& \text{\( \begin{aligned} 4x-6y+10z&=8\\ 6x-9y+15z&=12\\ x+y+z&=1\\ \end{aligned}\)} \\\hline \text{\(\begin{aligned} 4x-6y+10z&=8\\ -2x+3y+5z&=4\\ x+y+z&=1\\ \end{aligned}\)}& \text{\( \begin{aligned} x+y+z&=1 \\ 2x+2y+2z&=2 \\ -\frac x2-\frac y2-\frac z2&=-\frac12 \end{aligned}\)} \end{array} \]
\( 2 \)
\( 1 \)
\( 3 \)
\( 4 \)

1003060503

Level: 
B
The system of equations is given by: \[ \begin{aligned} x-y-z&=0, \\ 2x-y+3z&=1, \\ -3x+2y+z&=2. \end{aligned} \] To which of the following systems is it equivalent? (Note: An algorithm for solving a system of linear equations by transformation the system into this form (row echelon form) is known as Gaussian elimination or as row reduction.)
\( \begin{aligned} x-y-z&=0 \\ y+5z&=1 \\ 3z&=3 \end{aligned} \)
\( \begin{aligned} x-y-z&=0 \\ y+5z&=-1 \\ 3z&=-1 \end{aligned} \)
\( \begin{aligned} x-y-z&=0 \\ -3y-z&=-1 \\ 5z&=5 \end{aligned} \)
\( \begin{aligned} x-y-z&=0 \\ -3y-z&=-1 \\ 5z&=-7 \end{aligned} \)

1003060502

Level: 
B
The system of equations is given by: \[ \begin{aligned} x+y-2z&=0, \\ x+2y+3z&=0, \\ -2x+y+z&=2. \end{aligned} \] To which of the following systems is it equivalent? (Note: An algorithm for solving a system of linear equations by transformation the system into this form (row echelon form) is known as Gaussian elimination or as row reduction.)
\( \begin{aligned} x+y-2z&=0 \\ y+5z&=0 \\ 18z&=-2 \end{aligned} \)
\( \begin{aligned} x+y-2z&=0 \\ y-5z&=0 \\ 12z&=2 \end{aligned} \)
\( \begin{aligned} x+y-2z&=0 \\ y+5z&=0 \\ 18z&=2 \end{aligned} \)
\( \begin{aligned} x+y-2z&=0 \\ y+z&=0 \\ 6z&=2 \end{aligned} \)