Systems of nonlinear equations and inequalities

1003160801

Level:

Use the substitution method to find the solution

[x; y]

of the following system of equations.

\begin{array}{r} \frac{2}{x + 4} - \frac{1}{2 - y} = - 6 \\ \frac{1}{x + 4} + \frac{5}{2 - y} = 8 \end{array}

[- \frac{9}{2}; \frac{3}{2}]

[- 2; 2]

[2; 10]

[- \frac{9}{2}; 3]

1003160802

Level:

Use the substitution method to find the solution

[x; y]

of the following system of equations.

\begin{array}{r} \frac{2 x}{x + 3} - 3 \cdot \frac{y + 2}{y} = 2 \\ \frac{x}{x + 3} + 2 \cdot \frac{y + 2}{y} = 8 \end{array}

[- 4; 2]

[4; 2]

[2; - 4]

[- 1; 2]

1003160803

Level:

Use the substitution method to find the solution

[x; y]

of the following system of equations.

\begin{array}{r} \frac{x + y}{x} + \frac{1}{x + y} = 1 \\ \frac{2 \cdot (x + y)}{x} - \frac{1}{x + y} = - 7 \end{array}

[- \frac{1}{6}; \frac{1}{2}]

[- 2; 3]

[- \frac{1}{2}; - \frac{1}{2}]

[\frac{1}{2}; \frac{3}{- 2}]

1103085403

Level:

Two resistors of unknown resistances

R_{1}

and

R_{2}

, where

R_{1} < R_{2}

are connected in series (Figure A) and total resistance of the circuit is

R_{S} = 64 Ω

. If the resistors are connected in parallel (Figure B), the total resistance

R_{P} = 15 Ω

. Find

R_{1}

24

22

12

15

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

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.

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}}

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}}

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.

9000020906

Level:

Identify an equation which can be obtained from the following system by eliminating one of the variables.

\begin{aligned} y^{2} & - & 2 & x & + & 3 & = 0 \\ x & - & y & - & 1 & = 0 \end{aligned}

(y - 1)^{2} = 0

(y + 1)^{2} = 0

(x - 4)^{2} = 0

(x + 2)^{2} = 0

Systems of nonlinear equations and inequalities

1003160801

1003160802

1003160803

1103085403

2000020302

2000020303

2000020305

2000020307

2100020306

9000020906

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