Systems of linear equations and inequalities

9000023908

Level: 
A
Let \([x;y]\) be the solution of the system \[\begin{aligned} 2x - y & = -1, & & \\4x - y & = 1. & & \end{aligned}\] In the following list identify a true statement.
\(y\) is a prime number.
\(x\) is a prime number.
\(x + y\) is a prime number.
\(x - y\) is a prime number.

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?
\( 21 \)
\( 39 \)
\( 20 \)
\( 15 \)

1003034505

Level: 
B
The March price of a T-shirt and shorts was \( 600\,\mathrm{CZK} \) together. In April there was on store price adjustment. The price of the shorts decreased by \( 10\% \) and the price of the T-shirt increased by \( 10\% \). So the April price of both together the shorts and the T-shirt was by \( 20\,\mathrm{CZK} \) lower. What was the April price of the T-shirt?
\( 220\,\mathrm{CZK} \)
\( 200\,\mathrm{CZK} \)
\( 180\,\mathrm{CZK} \)
\( 400\,\mathrm{CZK} \)

1003034506

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 \( 8 \) 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 \( 9 \) hours. How long will they mow together?
\( 2 \) hours
\( 7 \) hours
\( 6 \) hours
\( 3 \) hours

1003060501

Level: 
B
Which of the following systems of equations has no solution?
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=4 \\ x+y+z&=1 \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=6 \\ x+y+z&=1 \\ \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-2y+z&=4 \\ x+y+z&=1 \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=6 \\ 4x-4y+6z&=4 \end{aligned} \)

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

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