Systems of linear equations and inequalities

1003060501

Level: 
B
Which of the following systems of equations has no solution?
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=4 \\ x+y+z&=1 \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=6 \\ x+y+z&=1 \\ \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-2y+z&=4 \\ x+y+z&=1 \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=6 \\ 4x-4y+6z&=4 \end{aligned} \)

1103020106

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given inequalities. Which inequality is it?
\( y > \frac32x+\frac{13}2 \)
\( y \geq \frac32x+\frac{13}2 \)
\( y < \frac32x+\frac{13}2\)
\( y \leq \frac32x+\frac{13}2\)

1003020304

Level: 
A
Find the solution set of the equation \[1-\left[4x+3\cdot(x-y)\right]=\frac{1-14x}2-\frac{3-6y}2\] in \( \mathbb{R}\times\mathbb{R} \).
\( \emptyset \)
\( \left\{[x;y],x\in\mathbb{R},y\in\mathbb{R}\right\} \)
\( \left\{\left[x;\frac13+x\right],x\in\mathbb{R}\right\} \)
\( \left\{\left[x;-\frac13\right],x\in\mathbb{R}\right\} \)

1003020303

Level: 
A
Find the solution set of the equation \[1-\frac{x-2y}4=x+\frac{y+2}2\] in \( \mathbb{R}\times\mathbb{R} \).
\( \left\{[0;y],y\in\mathbb{R}\right\} \)
\( \left\{[x;y],x\in\mathbb{R},y\in\mathbb{R}\right\} \)
\( \left\{\left[\frac{-4y}5;y\right],y\in\mathbb{R}\right\} \)
\( \{0\} \)
\( \emptyset \)

1003020302

Level: 
A
Assuming \( [x;y]\in\mathbb{R}\times\mathbb{R} \), solve the equation \[ x-y-\frac{x-y}2=\frac{x-y}3 \] Decide which of the answers below does not express the set of roots.
\( \left\{[x;y],x\in\mathbb{R},y\in\mathbb{R}\right\} \)
\( \left\{[x;x],x\in\mathbb{R}\right\} \)
\( \left\{[y;y],y\in\mathbb{R}\right\} \)
\( \left\{[t;t],t\in\mathbb{R}\right\} \)