Systems of linear equations and inequalities

2000019205

Level: 
B
Let \([x, y, z]\) be the solution of the system of \(3\) equations with \(3\) unknowns represented by the matrix \[\left(\begin{array}{ccc|c} 1 & 2 & 1 & 6 \\ 2 & -1 & 1 & 1\\ -1 & 1 & 1 & 2 \end{array}\right). \] Which of the components \(x\), \(y\), and \(z\) is of the highest value?
\(y\)
\(x\)
\(z\)
cannot be identified

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

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

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?
\(40\)
\(58\)
\(13\)
\(50\)

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?
\[\begin{aligned} x +5y + 7z & = 198 & & \\ x - y+z & = 6 & & \\ x +y+z & = 40 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\(x+6) +5(y+6)+7(z+6) & = 276 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x + y-z & = 6 & & \\(x+6) +5(y+6)+7(z+6) & = 276 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 202 & & \\x - y+z & = 6 & & \\(x+6) +(y+6)+(z+6) & = 58 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\x +5y+7z & = 40 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\(x-6) +5(y-6)+7(z-6) & = 276 & & \end{aligned}\]

2000019004

Level: 
B
The system of equations is given by: \[\begin{aligned} 2 x-y +z=5 & & \\x +2y-3z =17& & \\x +y -2z= 12& & \end{aligned}\] When solving the system using Cramer's rule, we evaluate determinants of four matrices. What is the sum of all these determinants?
\(-14\)
\(12\)
\(0\)
\(-20\)

2000019007

Level: 
B
The system of equations is given by: \[\begin{aligned} x+2z= 3 & & \\2x -y+ z = 2& & \\3x -2 y -z= 1 & & \end{aligned}\] When solving the system using Cramer's rule, we evaluate determinants of four matrices. What is the arithmetic mean of all these determinants?
\(2 \)
\(3.5 \)
\(\frac73 \)
\(\frac83 \)