Systems of linear equations and inequalities

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 \( 7 \)

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 \( 10\,800\,\mathrm{CZK} \)

1003060502

Level: 
C
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.)
\( \begin{aligned} x+y-2z&=0 \\ y+5z&=0 \\ 18z&=-2 \end{aligned} \)
\( \begin{aligned} x+y-2z&=0 \\ y-5z&=0 \\ 12z&=2 \end{aligned} \)
\( \begin{aligned} x+y-2z&=0 \\ y+5z&=0 \\ 18z&=2 \end{aligned} \)
\( \begin{aligned} x+y-2z&=0 \\ y+z&=0 \\ 6z&=2 \end{aligned} \)

1003060503

Level: 
C
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.)
\( \begin{aligned} x-y-z&=0 \\ y+5z&=1 \\ 3z&=3 \end{aligned} \)
\( \begin{aligned} x-y-z&=0 \\ y+5z&=-1 \\ 3z&=-1 \end{aligned} \)
\( \begin{aligned} x-y-z&=0 \\ -3y-z&=-1 \\ 5z&=5 \end{aligned} \)
\( \begin{aligned} x-y-z&=0 \\ -3y-z&=-1 \\ 5z&=-7 \end{aligned} \)

1003060504

Level: 
C
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} \]
\( 2 \)
\( 1 \)
\( 3 \)
\( 4 \)

1103020106

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?
\( y > \frac32x+\frac{13}2 \)
\( y \geq \frac32x+\frac{13}2 \)
\( y < \frac32x+\frac{13}2\)
\( y \leq \frac32x+\frac{13}2\)