Systems of Nonlinear Equations and Inequalities

9000031108

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} 2x^{2} - y = 2 & & \\\left |x\right | + y = 1 & & \end{aligned}\]
The system has two solutions.
The system does not have any solution.
The system has a unique solution.
The system has more than two solutions.

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.

9000031109

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \left |x\right | = x + y & & \\\left |y\right | = 1 + x & & \end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has more than two 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.

9000031102

Level: 
B
Solve the following system of equations and identify a correct statement. \[\begin{aligned} (x - 1)^{2} + y^{2} = 1 & & \\(x - 4)^{2} + y^{2} = 4 & & \end{aligned}\]
The system has a unique solution \(\left [x,y\right ]\), where \(y = 0\).
The system does not have any solution.
The system has a unique solution \(\left [x,y\right ]\), where \(y > 0\).
The system has two solutions \(\left [x_{1},y_{1}\right ]\), \(\left [x_{2},y_{2}\right ]\), where \(y_{1} = -y_{2}\).

9000031106

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \sqrt{x + y} & = \left |x\right | & & \\x + y & = 4 & & \end{aligned}\]
The system has two solutions \(\left [x_{1},y_{1}\right ]\), \(\left [x_{2},y_{2}\right ]\), where \(x_{1} = -x_{2}\).
The system does not have any solution.
The system has a unique solution.
The system has two solutions \(\left [x_{1},y_{1}\right ]\), \(\left [x_{2},y_{2}\right ]\), where \(x_{1} = x_{2}\).

9000031107

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \sqrt{x} = y & & \\x^{2} + y^{2} = 6 & & \end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has more than two solutions.

9000022907

Level: 
C
Solve the following system of equations and identify a true statement. \[ \begin{alignedat}{80} |x - 2| & + &y & = &2 & & & & & & \\ - 2|5 + x| &- 3 &y & = - &5 & & & & & & \\\end{alignedat}\]
The system has two solutions. Both solutions satisfy \(y < 0\).
The system has a unique solution. This solution satisfies \(y > 0\).
The system has two solutions. Both solutions satisfy \(y > 0\).
The system has more than two solutions.
The system does not have any solution.

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