Systems of Nonlinear Equations and Inequalities

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.

9000031110

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \left |x - 2\right | & = y & & \\\left |y + 2\right | & = x - 6 & & \end{aligned}\]
The system has no solution.
The system has a unique solution.
The system has two solutions.
The system has more than two solutions.

9000031101

Level: 
B
Solve the following system of equations and identify a correct statement. \[\begin{aligned} (x - 3)^{2} + (y - 1)^{2} = 1 & & \\2x^{2} + 2y^{2} - 12x - 4y + 18 = 0 & & \end{aligned}\]
The system has more than two solutions.
The system does not have any solution.
The system has a unique solution.
The system has two solutions.

9000031103

Level: 
B
Solve the following system of equations in \( \mathbb{R} \times \mathbb{R}\) and identify the correct statement.\[\begin{aligned} x - 2y + 5 = 0 & & \\x^{2} + y^{2} = 9 & & \end{aligned}\]
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.

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.

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.

9000020906

Level: 
A
Identify an equation which can be obtained from the following system by eliminating one of the variables. \[ \begin{alignedat}{80} &y^{2} & - &2 &x & + &3 & = 0 & & & & & & & & \\ &x & - & &y & - &1 & = 0 & & & & & & & & \\\end{alignedat}\]
\((y - 1)^{2} = 0\)
\((y + 1)^{2} = 0\)
\((x - 4)^{2} = 0\)
\((x + 2)^{2} = 0\)

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.