Systems of linear equations and inequalities

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?
\(2 \)
\(3.5 \)
\(\frac73 \)
\(\frac83 \)

2000019006

Level: 
B
The coefficient matrix of a system of three linear equations with three unknowns is: \[ \left (\array{ 1& 2& 1\cr 3& -5& 2\cr 1& 0& -3} \right ).~ \] What is the column of the right sides if the solution is the ordered triple \([−7; 2;−1]\)?
\( \left (\array{ -4\cr -33\cr -4} \right ) \)
\( \left (\array{ -2\cr -33\cr -4} \right ) \)
\( \left (\array{ -4\cr -31\cr -4} \right ) \)
\( \left (\array{ -4\cr -33\cr -10} \right ) \)

2000019005

Level: 
B
To solve the system of three linear equations with three unknowns, it is necessary to calculate the determinants of the matrices: \[ \left (\array{ 1& -2& 3\cr 2& 1& -7\cr -3& 1& -5} \right ),~ \left (\array{ 1& 3& -1\cr 2& -7& -3\cr -3& -5& 1} \right ). \] Which of the given ordered triples is the solution to this system?
\( [2,-2,3]\)
\( [2,2,3]\)
\( [-2,2,3]\)
\( [3,-2,2]\)

2000019003

Level: 
B
Consider a linear system of three equations with three unknowns \(x\), \(y\), \(z\), and with the column of the right sides: \[ \left (\array{ 5\cr 17\cr 12} \right ) \] The determinants of the following two matrices were used to solve the system by Cramer's rule: \[ \left (\array{ 2& 5& 1\cr 1& 17& -3\cr 1& 12& -2} \right ),~ \left (\array{ 2& -1& 5\cr 1& 2& 17\cr 1& 1& 12} \right ) \] Which of the following systems could be solved in a specified way?
\[\begin{aligned} 2x- y +z= 5 & & \\x +2y-3 z = 17 & & \\x + y -2z= 12 & & \end{aligned}\]
\[\begin{aligned} 2x+5 y +z= -1 & & \\x +17y-3 z = 2& & \\x +12 y -2z= 1 & & \end{aligned}\]
\[\begin{aligned} 2x- y +z= -5 & & \\x +2y-3 z = -17 & & \\ x+y -2z= -12& & \end{aligned}\]
\[\begin{aligned} 2x+ y-z = 5 & & \\x-2y + 3z = 17 & & \\x - y +2z= 12 & & \end{aligned}\]

2000019002

Level: 
B
The system of equations is given by: \[\begin{aligned} x- y = -3 & & \\2x + z = -5 & & \\x + y -z= 0 & & \end{aligned}\] When solving the system using Cramer's rule, we evaluate determinants of four matrices. Suppose we order them according to their values. What is the largest value of these determinants?
\(8\)
\(4\)
\(-4\)
\(12\)