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

2000019206

Level: 
B
For what value of a real number \(a\) has the following system infinitely many solutions? \[ \begin{alignedat}{80} &x & + &2y & +& z & = 8 & & & & & & \\ &2x & & & -& z & = -1 & & & & & & \\ &7x & + & 10y & +& 4z & = a & & & & & & \\\end{alignedat}\]
\(39\)
\(73\)
\(-39\)
\(56\)

2000019207

Level: 
B
Adam went to a shop where he bought \(7\) buns and \(2\) cakes for \(64\) Kč. Mirek bought \(5\) buns, \(3\) cakes and \(4\) rolls for \(79\) Kč. Petra went to the same shop as Adam and Mirek and bought \(5\) buns and \(4\) rolls. Since it was only \(20\) minutes to closing, she got a discount for each piece of bakery goods worth \(1\) Kč and so she paid \(37\) Kč. Which of the following statements about the prices of goods before the discount is the only incorrect?
\(2\) buns and \(1\) cake altogether cost more than \(16\) rolls
The cake is more expensive than a bun and a roll altogether.
\(3\) cakes cost more than \(8\) rolls.
Buying \(10\) pieces of each (a bun, cake, and a roll) cost more than \(200\) Kč.

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?
\(\begin{aligned} 5x+8y& \leq 27 \\ 9x+2y &< -15 \end{aligned}\)
\(\begin{aligned} 5x+8y &< 27 \\ 9x+2y &\leq -15 \end{aligned}\)
\(\begin{aligned} 5x+8y &\geq 27\\ 9x+2y &> -15 \end{aligned}\)
\(\begin{aligned} 5x+8y &> 27 \\ 9x+2y &\geq -15 \end{aligned}\)

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?
\(\begin{aligned} 3x-4y &>6\\ -1.5x+2y &< 5 \end{aligned}\)
\(\begin{aligned} 3x-4y &< 6\\ -1.5x+2y& < 5 \end{aligned}\)
\(\begin{aligned} 3x-4y &< 6\\ -1.5x+2y &> 5 \end{aligned}\)
\(\begin{aligned} 3x-4y &> 6\\ -1.5x+2y& > 5 \end{aligned}\)

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?
\(\begin{aligned} \frac{x}3-\frac{x}4 &> 2x-\frac12 \\ 3x+8 &\geq 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4 &\geq 2x-\frac12\\ 3x+8 &> 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4& < 2x-\frac12 \\ 3x+8 &\geq 2-\frac52x \end{aligned}\)
\(\begin{aligned} \frac{x}3-\frac{x}4 &> 2x-\frac12 \\ 3x+8 &\leq 2-\frac52x \end{aligned}\)

2000017706

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