Systems of linear equations and inequalities

2000020401

Level: 
A
The system of two linear equations can be represented graphically by two lines. Decide which of the systems given below corresponds to the following picture.
\[\begin{aligned} x-y&=-4\\ x+\frac53y&=-\frac43\\ \end{aligned}\]
\[\begin{aligned} x-y&=-4\\ \frac13x+\frac53y&=-\frac43\\ \end{aligned}\]
\[\begin{aligned} x+y&=-4\\ x+\frac53y&=-\frac43\\ \end{aligned}\]
\[\begin{aligned} x-y&=-4\\ 3x+5y&=-\frac43\\ \end{aligned}\]

2000020403

Level: 
A
In a system of two linear equations with two unknowns, the assignment of the second equation is inadvertently blurred, but we know that the first component of the solution of the system is \(x=-1\). We do not know the value of \(y\), but the part of the figure illustrating the graphical solution is preserved. The first equation is \(x-y+2=0\). Determine the second (blurred) equation of this system.
\(7x-11y+18=0\)
\(x-y+2=0\)
\(7x+11y-18=0\)
\(x+y+2=0\)

2000020406

Level: 
A
Let's denote by \(M\) the set of all points in the plane such that their coordinates \(\left[x;y\right]\) satisfy the relation \(2x-y+1=0\). Then, choose the true statement about \(M\).
\(M\) is a line.
\(M\) is a ray.
\(M\) is a finite set of point.
\(M\) is a half plane.

2000019201

Level: 
B
People in Kocourkov pay by coins named groschen. Their coins are worth \(1\), \(5\) or \(7\) groschen. Martin and Petr, who live in Kocourkov, emptied their saving boxes and started counting their saved coins. They found out that Petr had \(6\) pieces of each type of a coin more than Martin, who had \(40\) coins in total. They were surprised to find out that Martin has the same number of \(1\)-groschen and \(7\)-groschen coins in total as Petr has of \(5\)-groschen ones. Petr was proud to have \(78\) groschen more than Martin, who was only \(2\) short of having \(200\) groschen. Which of the following systems can be used to find out how many coins of each type both boys have?
\[\begin{aligned} x +5y + 7z & = 198 & & \\ x - y+z & = 6 & & \\ x +y+z & = 40 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\(x+6) +5(y+6)+7(z+6) & = 276 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x + y-z & = 6 & & \\(x+6) +5(y+6)+7(z+6) & = 276 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 202 & & \\x - y+z & = 6 & & \\(x+6) +(y+6)+(z+6) & = 58 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\x +5y+7z & = 40 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\(x-6) +5(y-6)+7(z-6) & = 276 & & \end{aligned}\]

2000019202

Level: 
B
People in Kocourkov pay by coins named groschen. Their coins are worth \(1\), \(5\) or \(7\) groschen. Martin and Petr, who live in Kocourkov, emptied their saving boxes and started counting their saved coins. They found out that Petr had \(6\) pieces of each type of a coin more than Martin, who had \(40\) coins in total. They were surprised to find out that Martin has the same number of \(1\)-groschen and \(7\)-groschen coins in total as Petr has of \(5\)-groschen ones. Petr was proud to have \(78\) groschen more than Martin, who was only \(2\) short of having \(200\) groschen. How many coins did Martin have?
\(40\)
\(58\)
\(13\)
\(50\)

2000019203

Level: 
B
The sweet shop offers \(3\) types of confections in various packages. The price of each package can be seen below the package (as shown in the picture). How much would the sample package cost if it contained \(1\) piece of each type of confection?
\(35\) ¢
\(30\) ¢
\(34\) ¢
none of the given prices

2000019204

Level: 
B
Visitors of a ZOO can buy a package with bags of goat food (blue color), sheep food (red color) and duck food (green color). The feed bags are offered in \(3\) various packages and their prices can be seen below the packages (as shown in the picture). Which of the feed is the most expensive one?
sheep food
goat food
duck food
cannot be identified