Systems of Linear Equations and Inequalities

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.
\( \left\{[x,y],x\in\mathbb{R},y\in\mathbb{R}\right\} \)
\( \left\{[x,x],x\in\mathbb{R}\right\} \)
\( \left\{[y,y],y\in\mathbb{R}\right\} \)
\( \left\{[t,t],t\in\mathbb{R}\right\} \)

1003020301

Level: 
A
In \( \mathbb{R}\times\mathbb{R} \), find the solution set of the equation: \[ 2x-\frac{x+2y}3=2+\frac83y \]
\( \left\{\left[2y+\frac65,y\right],y\in\mathbb{R}\right\} \)
\( \left\{\left[2y+\frac65,\frac x2-\frac35\right],x\in\mathbb{R},y\in\mathbb{R}\right\} \)
\( \left\{\left[\frac{6+6y}5,y\right],y\in\mathbb{R}\right\} \)
\( \emptyset \)

9000026006

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?
\(\begin{aligned}x +\phantom{ 2}y&\geq 3 & \\y - 2x& < -1 \\ \end{aligned}\)
\(\begin{aligned}x +\phantom{ 2}y& > 3 & \\y - 2x& < -1 \\ \end{aligned}\)
\(\begin{aligned}x +\phantom{ 2}y&\leq 3 & \\y - 2x& < -1 \\ \end{aligned}\)
\(\begin{aligned}x +\phantom{ 2}y& < 3 & \\y - 2x& > -1 \\ \end{aligned}\)