Systems of nonlinear equations and inequalities

9000031104

Level: 
A
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \frac{x} {y + 1} - \frac{2} {x + 1} & = 0 & & \\\frac{y} {x} + \frac{2} {x} & = -1 & & \end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has infinitely many solutions.

9000031105

Level: 
A
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \frac{1} {x + 1} -\frac{1} {y} = 0 & & \\y^{2} = 1 & & \end{aligned}\]
The system has two solutions.
The system does not have any solution.
The system has a unique solution.
The system has infinitely many solutions.

2000020304

Level: 
B
Solve the given system of equations in the set of real numbers. \[\begin{aligned} x-y&=2\\ x^2-y^2&=2\\ \end{aligned}\] In the following list identify a true statement.
The system has exactly one solution.
The system has no solution.
The system has infinitely many solutions.
The quotient of numbers \(x\) and \(y\) is \(3\).

2010006506

Level: 
B
Solve the following system of equations in \( \mathbb{R} \times \mathbb{R}\) and identify the correct statement. \[\begin{aligned} x^2 - y^2 = 5 & & \\2x + y = 1 & & \end{aligned}\]
The system does not have any solution.
The system has a unique solution.
The system has more than two solutions.
The system has two solutions.

2010006703

Level: 
B
Identify a true statement related to the solution of the following system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &2 & &y^{2} & - & &4x & = &0 & & & & & & & & & & & & \\ &x & + & & &y & = &4 & & & & & & & & & & & & \\\end{alignedat}\]
The system has two solutions.
The system has a unique solution.
The system does not have any solution.
The system has infinitely many solutions.

2010006704

Level: 
B
Determine all the values of the parameter \(c\in \mathbb{R}\) so that the following system has two solutions in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &2y^{2} & = 6 & & & & & & \\ &x & + &y & = c & & & & & & \\\end{alignedat}\]
\(|c| < 3\)
\(|c| =3\)
\(|c| > 3\)
\(|c| \in \mathbb{R}\)

9000020901

Level: 
B
The solution of the given set of equations can be interpreted as the intersection of the curves shown in the figure. Find the solution of the system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &2x^{2} & - &3y &^{2} & = 2 &4 & & & & & & & & \\ &2x & - &3y & & = &0 & & & & & & & & \\\end{alignedat}\]
\([-6;-4],\ [6;4]\)
\([-6;-4]\)
\([6;4]\)
no solution

9000020902

Level: 
B
The solution of the given set of equations can be interpreted as the intersection of the curves shown in the figure. Find the solution of the system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &4x^{2} & + &y &^{2} & = &20 & & & & & & & & & \\ &2x & + &y & & = &6 & & & & & & & & & \\\end{alignedat}\]
\([1;4],\ [2;2]\)
\([2;2]\)
\([1;4]\)
no solution

9000020903

Level: 
B
Identify a true statement related to the solution of the following system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &4 & &y^{2} & - & &2x & = &15 & & & & & & & & & & & & \\ &x & - & & &y & + & &1 & = &0 & & & & & & & & & & & & \\\end{alignedat}\]
The system has two solutions.
The system has a unique solution.
The system does not have any solution.
The system has infinitely many solutions.

9000020907

Level: 
B
Identify a true statement related to the solution of the following system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &2x^{2} & - & &y^{2} & - &2 &x & - 5 & = 0 & & & & & & & & & & \\ & & & &3x & - & &y & - 5 & = 0 & & & & & & & & & & \\\end{alignedat}\]
The system has no solution.
The system has two solutions.
The system has a unique solution.
None of the above conclusions can be obtained.