Systems of linear equations and inequalities

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?
\(8\)
\(4\)
\(-4\)
\(12\)

2000019001

Level: 
B
Four matrices are given: \[\] $\left (\array{ 1& -1& 0\cr 2& 0& 1\cr 1& 1& -1} \right ),$ $\left (\array{ 1& -3& 0\cr 2& -5& 1\cr 1& 0& -1} \right ),$ $\left (\array{ -3& -1& 0\cr -5& 0& 1\cr 0& 1& -1} \right ),$ $\left (\array{ 1& -1& -3\cr 2& 0& -5\cr 1& 1& 0} \right )$ \[\] We want to practice Cramer's rule for solving a system of linear equations. Which of the following systems can be solved using determinants of the four matrices given above?
\[\begin{aligned} x- y = -3 & & \\2x + z = -5 & & \\x + y -z= 0 & & \end{aligned}\]
\[\begin{aligned} x- y-3z = 0 & & \\2x - 5z = 1 & & \\x + y = -1& & \end{aligned}\]
\[\begin{aligned} -3x- y = 0 & & \\-5x + z = 1 & & \\ y -z= -1& & \end{aligned}\]
\[\begin{aligned} x- y = 3 & & \\2x + z = 5 & & \\x + y -z= 0 & & \end{aligned}\]

2000017706

Level: 
C
Which of the systems has its solution graphed on the number line?
\(\begin{aligned} -5x-4 &>11-2x \\ 8-9x &> 2x-69 \end{aligned}\)
\(\begin{aligned} -5x-4 &>11-2x \\ 8-9x& < 2x-69 \end{aligned}\)
\(\begin{aligned} -5x-4 &< 11-2x\\ 8-9x &< 2x-69 \end{aligned}\)
\(\begin{aligned} -5x-4& < 11-2x\\ 8-9x &> 2x-69 \end{aligned}\)

2000017705

Level: 
C
The interval \( \left[ -\frac{12}{11}; \frac6{23}\right)\) is the solution of a system of two linear inequalities with one unknown. Which of the following systems is it?
\(\begin{aligned} \frac{x}3-\frac{x}4 &> 2x-\frac12 \\ 3x+8 &\geq 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4 &\geq 2x-\frac12\\ 3x+8 &> 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4& < 2x-\frac12 \\ 3x+8 &\geq 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4 &> 2x-\frac12 \\ 3x+8 &\leq 2-\frac52x \end{aligned}\)

2000017703

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?
\(\begin{aligned} 3x-4y &>6\\ -1.5x+2y &< 5 \end{aligned}\)
\(\begin{aligned} 3x-4y &< 6\\ -1.5x+2y& < 5 \end{aligned}\)
\(\begin{aligned} 3x-4y &< 6\\ -1.5x+2y &> 5 \end{aligned}\)
\(\begin{aligned} 3x-4y &> 6\\ -1.5x+2y& > 5 \end{aligned}\)

2000017702

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?
\(\begin{aligned} 5x+8y& \leq 27 \\ 9x+2y &< -15 \end{aligned}\)
\(\begin{aligned} 5x+8y &< 27 \\ 9x+2y &\leq -15 \end{aligned}\)
\(\begin{aligned} 5x+8y &\geq 27\\ 9x+2y &> -15 \end{aligned}\)
\(\begin{aligned} 5x+8y &> 27 \\ 9x+2y &\geq -15 \end{aligned}\)