Systems of nonlinear equations and inequalities

2000020307

Level: 
A
Describe the set of all ordered pairs of real numbers in the form \([x;y]\) that are solutions to the followig equation. \[ \frac{x-7}{y+1}=5 \] Which of the descriptions of our solution set is correct?
\[ \left\{ \left[5m+12;m\right];m\in\mathbb{R}\setminus \left\{-1\right\}\right\} \]
\[ \left\{ \left[x;0.2x-2.4\right];x\in\mathbb{R}\setminus \left\{-0.7\right\}\right\} \]
\[ \left\{ \left[5a-12;a\right];a\in\mathbb{R}\setminus \left\{-1\right\}\right\} \]
\[ \left\{ \left[q;0.2q+2.4\right];q\in\mathbb{R}\setminus \left\{-1.8\right\}\right\} \]

2000020305

Level: 
A
Describe the set of all ordered pairs of real numbers in the form \(\left[x;y\right] \) that are solutions to the following equation. \[\frac{y+2}{x-4}=3\] Which of the descriptions of our solution set is incorrect?
\[ \left\{ \left[2b;b+\frac{14}{3}\right];b\in\mathbb{R}\setminus \left\{2\right\}\right\} \]
\[ \left\{ \left[x;3x-14\right];x\in\mathbb{R}\setminus \left\{4\right\}\right\} \]
\[ \left\{ \left[\frac{y+14}{3};y\right];y\in\mathbb{R}\setminus \left\{-2\right\}\right\} \]
\[ \left\{ \left[\frac{a}{3};a-14\right];a\in\mathbb{R}\setminus \left\{12\right\}\right\} \]

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 \(3\).

2000020303

Level: 
A
Solve the given system of equations in the set of real numbers. \[\begin{aligned} x+y&=4+\frac{1}{27}\\ x\cdot y&=\frac{4}{27}\\ \end{aligned}\] In the following list identify a true statement.
\(|x-y|=\frac{107}{27}\)
The system has exactly one solution.
The system has no solution.
The system has infinitely many solutions.

2000020302

Level: 
A
Solve the given system of equations in the set of real numbers. \[\begin{aligned} x^2+y&=2\\ 2x-y+3&=0\\ \end{aligned} \] In the following list identify a true statement.
The numbers \(x\) and \(y\) are opposite of each other.
The sum of the numbers \(x\) and \(y\) is equal to \(-2\).
The arithmetic mean of numbers \(x\) and \(y\) is equal to \(2\).
The ratio of numbers \(x\) and \(y\) is \(2:1\).

2000020301

Level: 
C
Solve the given system of equations in the set of real numbers. \[ \begin{aligned} x+y&=-5\\ 1+\sqrt{2x+4y}&=\sqrt{x+3y}\\ \end{aligned}\] In the following list identify a true statement.
\(x=-12,\ y=7\)
\(x=12,\ y=7\)
The system has no solution.
The system has infinitely many solutions.

2000017704

Level: 
C
Assuming \( x \in \mathbb{R}\), find the solution set of the following system of inequalities. \[\begin{aligned} 2x- [x-(2x+1)]\cdot 3 &> (3+x)-2(1-x)-2x+6 \\ x^2-3\cdot [x-2x(1-x)] &< 5(10-x^2)-2x \end{aligned}\]
\( (1;10)\)
\( \emptyset \)
\( (-10;1)\)
\( \{1;10\}\)

2010011206

Level: 
C
Consider the system \[\begin{aligned} y & = \frac{k} {x}, & & \\y & = a, & & \end{aligned}\] where \(a\), \(k\) are real parameters and \(x\), \(y\) are real variables. Determine the conditions for \(a\) and \(k\) so that the system has a unique solution in \(\mathbb{R}^{+}\times \mathbb{R}^{-}\).
\(a < 0\) and \(k < 0\)
\(a < 0\) and \(k > 0\)
\(a > 0\) and \(k < 0\)
\(a > 0\) and \(k > 0\)