Systems of linear equations and inequalities
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.
2100020405
Level:
A
Provided that the missing parts of lines were completed, which of the pictures would depict solution of the system if solved graphically?
\[\begin{aligned}
x-7y&=-9\\
3x-2y&=11\\
\end{aligned}\]
2100020404
Level:
A
The system of two linear equations can be represented graphically by two lines. Decide which of the pictures corresponds the following system.
\[\begin{aligned}
-x+3y&=3.5\\
1.4x+y&=2.9\\
\end{aligned}
\]
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.
\(7x-11y+18=0\)
\(x-y+2=0\)
\(7x+11y-18=0\)
\(x+y+2=0\)
2000020402
Level:
A
What is the solution of a system of two linear equations with two unknowns if one of the equations is represented by the blue line and the other by the red line (see the picture)?
\(x=6,\ y=-2\)
\(x=5,\ y=-2\)
\(x=6,\ y=-1\)
\(x=6,\ y=-1.5\)
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.
\[\begin{aligned}
x-y&=-4\\
x+\frac53y&=-\frac43\\
\end{aligned}\]
\[\begin{aligned}
x-y&=-4\\
\frac13x+\frac53y&=-\frac43\\
\end{aligned}\]
\[\begin{aligned}
x+y&=-4\\
x+\frac53y&=-\frac43\\
\end{aligned}\]
\[\begin{aligned}
x-y&=-4\\
3x+5y&=-\frac43\\
\end{aligned}\]
2000019208
Level:
B
Let \([x, y, z]\) be the solution of the system
\[\begin{aligned}
x +2 y & = \frac74 & &
\\y +3z & = 2.5 & &
\\4x +z & = \frac{11}3 & &
\end{aligned}\]
What is the value of \(x+y+z\)?
\(\frac{23}{12}\)
\(2\)
\(\frac{20}{12}\)
\(-\frac{23}{12}\)
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.
2000019206
Level:
B
For what value of a real number \(a\) has the following system infinitely many solutions?
\[ \begin{alignedat}{80}
&x & + &2y & +& z & = 8 & & & & & &
\\ &2x & & & -& z & = -1 & & & & & &
\\ &7x & + & 10y & +& 4z & = a & & & & & &
\\\end{alignedat}\]
\(39\)
\(73\)
\(-39\)
\(56\)