Systems of nonlinear equations and inequalities

1003160801

Level: 
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations. \[ \begin{aligned} \frac2{x+4}-\frac1{2-y}=-6 \\ \frac1{x+4}+\frac5{2-y}=8 \end{aligned} \]
\( \left[-\frac92;\frac32\right] \)
\( [-2;2] \)
\( [2;10] \)
\( \left[-\frac92;3\right] \)

1003160803

Level: 
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations. \[ \begin{aligned} \frac{x+y}x+\frac1{x+y}=1 \\ \frac{2\cdot(x+y)}x-\frac1{x+y}=-7 \end{aligned} \]
\( \left[-\frac16;\frac12\right] \)
\( [-2;3] \)
\( \left[-\frac12;-\frac12\right] \)
\( \left[\frac12;\frac3{-2}\right] \)

1103085403

Level: 
A
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 \( R_S=64\,\Omega \). If the resistors are connected in parallel (Figure B), the total resistance \( R_P=15\,\Omega \). Find \( R_1 \).
\( 24 \)
\( 22 \)
\( 12 \)
\( 15 \)

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 \(2:1\).

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.

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?
\[ \left\{ \left[2b;b+\frac{14}{3}\right];b\in\mathbb{R}\setminus \left\{2\right\}\right\} \]
\[ \left\{ \left[x;3x-14\right];x\in\mathbb{R}\setminus \left\{4\right\}\right\} \]
\[ \left\{ \left[\frac{y+14}{3};y\right];y\in\mathbb{R}\setminus \left\{-2\right\}\right\} \]
\[ \left\{ \left[\frac{a}{3};a-14\right];a\in\mathbb{R}\setminus \left\{12\right\}\right\} \]

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?
\[ \left\{ \left[5m+12;m\right];m\in\mathbb{R}\setminus \left\{-1\right\}\right\} \]
\[ \left\{ \left[x;0.2x-2.4\right];x\in\mathbb{R}\setminus \left\{-0.7\right\}\right\} \]
\[ \left\{ \left[5a-12;a\right];a\in\mathbb{R}\setminus \left\{-1\right\}\right\} \]
\[ \left\{ \left[q;0.2q+2.4\right];q\in\mathbb{R}\setminus \left\{-1.8\right\}\right\} \]

9000020906

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