Systems of linear equations and inequalities

2000004004

Level: 
A
Identify which of the following systems of equations has no solution.
\[\begin{aligned} y & = 10-2x & & \\ y & = 5 -2x& & \end{aligned}\]
\[\begin{aligned} y & = 10-2x & & \\ 2y & = 20 -4x& & \end{aligned}\]
\[\begin{aligned} y & = 10-2x & & \\ -y & = 5 -2x& & \end{aligned}\]
\[\begin{aligned} y & = 10-2x & & \\ 3y & = 30 -6x& & \end{aligned}\]

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.
\[\begin{aligned} x-y&=-4\\ x+\frac53y&=-\frac43\\ \end{aligned}\]
\[\begin{aligned} x-y&=-4\\ \frac13x+\frac53y&=-\frac43\\ \end{aligned}\]
\[\begin{aligned} x+y&=-4\\ x+\frac53y&=-\frac43\\ \end{aligned}\]
\[\begin{aligned} x-y&=-4\\ 3x+5y&=-\frac43\\ \end{aligned}\]

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.
\(7x-11y+18=0\)
\(x-y+2=0\)
\(7x+11y-18=0\)
\(x+y+2=0\)

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.

2010006501

Level: 
A
In \( \mathbb{R}\times\mathbb{R} \), find the solution set of the equation: \[ 3y-\frac{x+y}2=1-\frac43x \]
\( \left\{ \left[-3y+\frac65;y\right],\ y\in\mathbb{R}\right \} \)
\( \left\{ \left[-3y+\frac65;x+\frac13\right],\ x\in\mathbb{R},y\in\mathbb{R}\right \} \)
\( \left\{ \left[\frac13 y+\frac65;y\right],\ y\in\mathbb{R}\right \} \)
\( \emptyset \)

2010006502

Level: 
A
Identify which of the following systems of equations has infinitely many solutions.
\( \begin{aligned} \frac12x-3y&=12\\ -\frac{1}3x+2y&=-8 \end{aligned} \)
\( \begin{aligned} \frac13 x-2y&=12 \\ -\frac12 x+3y&=-16 \end{aligned} \)
\( \begin{aligned} \frac12 x+2y&=12 \\ -\frac13 x-3y&=-12 \end{aligned} \)
\( \begin{aligned} \frac12 x-y&=12 \\ -\frac23 x+4y&=-8\end{aligned} \)

2010006503

Level: 
A
Consider the linear system: \[ \begin{aligned}6x - 3y - 42& = 0,& \\\text{???}\quad & = 0. \\ \end{aligned} \] In the following list, identify the missing second equation if you know that the system does not have a solution.
\(- 2x + y +12 = 0\)
\( 2x + y +21 = 0\)
\(3x -2y -12 = 0\)
\(12x -6 y -84 = 0\)