Systems of nonlinear equations and inequalities

System of a Linear and a Quadratic Equation

Submitted by michaela.bailova on Fri, 12/06/2024 - 14:28

2100020306

Level:

We are given the equation

x^{2} - 6 x - y = - 7

, where

x

and

y

are real numbers. Choose the picture showing (in red color) the solution set of the equation.

2000020307

Level:

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?

{[5 m + 12; m]; m \in R ∖ {- 1}}

{[x; 0.2 x - 2.4]; x \in R ∖ {- 0.7}}

{[5 a - 12; a]; a \in R ∖ {- 1}}

{[q; 0.2 q + 2.4]; q \in R ∖ {- 1.8}}

2000020305

Level:

Describe the set of all ordered pairs of real numbers in the form

[x; y]

that are solutions to the following equation.

\frac{y + 2}{x - 4} = 3

Which of the descriptions of our solution set is incorrect?

{[2 b; b + \frac{14}{3}]; b \in R ∖ {2}}

{[x; 3 x - 14]; x \in R ∖ {4}}

{[\frac{y + 14}{3}; y]; y \in R ∖ {- 2}}

{[\frac{a}{3}; a - 14]; a \in R ∖ {12}}

2000020304

Level:

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

3

2000020303

Level:

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:

Solve the given system of equations in the set of real numbers.

\begin{aligned} x^{2} + y & = 2 \\ 2 x - 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

2 : 1

2000020301

Level:

Solve the given system of equations in the set of real numbers.

\begin{aligned} x + y & = - 5 \\ 1 + \sqrt{2 x + 4 y} & = \sqrt{x + 3 y} \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:

Assuming

x \in R

, find the solution set of the following system of inequalities.

\begin{aligned} 2 x - [x - (2 x + 1)] \cdot 3 & > (3 + x) - 2 (1 - x) - 2 x + 6 \\ x^{2} - 3 \cdot [x - 2 x (1 - x)] & < 5 (10 - x^{2}) - 2 x \end{aligned}

(1; 10)

\emptyset

(- 10; 1)

{1; 10}

2010011206

Level:

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

R^{+} \times R^{-}

a < 0

and

k < 0

a < 0

and

k > 0

a > 0

and

k < 0

a > 0

and

k > 0

Systems of nonlinear equations and inequalities

System of a Linear and a Quadratic Equation

2100020306

2000020307

2000020305

2000020304

2000020303

2000020302

2000020301

2000017704

2010011206

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