Systems of linear equations and inequalities

9000019910

Level: 
B
The augmented matrix of a system of three equations with three unknowns is row equivalent to the following matrix \(A'\). Find the solution of the system. \[ A' = \left(\begin{array}{ccc|c} -1 & -6 & 1 &-20\\ 0 & 5 & 4 & -12\\ 0 & 0 & 0 & -8 \end{array}\right) \]
no solution
\(\left [-\frac{172} {5} ;-\frac{12} {5} ;0\right ]\)
\([-12t;4t;-8t],\ t\in \mathbb{R}\)
\(\left [-12;4;-8\right ]\)

9000022904

Level: 
B
Find the values of a real parameter \(t\) which ensure that the following system has a unique solution. \[ \begin{alignedat}{80} 2x & + &y & + &t & = - &2 & & & & & & & & \\ - 4x & - 2 &y & + &1 & = &0 & & & & & & & & \\\end{alignedat}\]
\(t\in \emptyset \)
\(t\in \mathbb{R}\)
\(t = 3\)
\(t = 1\)
\(t\in \mathbb{R}\setminus \{3\}\)

9000022905

Level: 
B
Find the values of a real parameter \(t\) which ensure that the following system has a unique solution. \[ \begin{alignedat}{80} tx & + &y & + &3 & = 0 & & & & & & \\4x & - 2 &y & + &1 & = 0 & & & & & & \\\end{alignedat}\]
\(t\in \mathbb{R}\setminus \{ - 2\}\)
\(t\in \mathbb{R}\)
\(t = -2\)
\(t\in \emptyset \)

9000022906

Level: 
B
Find the values of a real parameter \(t\) which ensure that the following system has a unique solution \([a,b]\) such that both \(a\) and \(b\) are positive real numbers. \[ \begin{alignedat}{80} a & - &tb & = - &2 & & & & & & \\a & + 2 &tb & = &0 & & & & & & \\\end{alignedat}\]
\(t\in \emptyset \)
\(t\in \mathbb{R}^{+}\)
\(t\in \mathbb{R}^{-}\)
\(t = 0\)
\(t\in \mathbb{R}\)

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