Systems of Equations and Inequalities

In Problems 5–12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression 

7. $\frac{x^2+5}{x^2-4}$

Theater Revenues 

A movie theater charges $\backslash (8.00$ for adults, $\backslash )4.50$ for children, and $\backslash (6.00$ for senior citizens. One day the theater sold $405$ tickets and collected $\backslash )2320$ in receipts. Twice as many children’s tickets were sold as adult tickets. How many adults, children, and senior citizens went to the theater that day?

Solve the equation:

$3x+4=8-x$

(a) Graph the line : $3x+4y=12$.

(b) What is the slope of a line parallel to this line?

If a system of equations has no solution, it is said to be _______.

If a system of equations has one solution, the system is ________ and the equations are ________.

If the solution to a system of two linear equations containing two unknowns is $x=3, y=-2$, then the lines intersect at the point _________.

If the lines that make up a system of two linear equations are coincident, then the system is __________ and the equations are ___________.

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}2x-y=5\\ 5x+2y=8\end{array}\right. \\ x=2, y=-1; (2,-1) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}3x+2y=2\\ x-7y=-30\end{array}\right. \\ x=-2, y=4;(-2,4) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}3x-4y=4\\ \frac{1}{2}x-3y=-\frac{1}{2}\end{array}\right. \\ x=2, y=\frac{1}{2};\left(2,\frac{1}{2}\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}2x+\frac{1}{2}y=0\\ 3x-4y=-\frac{19}{2}\end{array}\right. \\ x=-\frac{1}{2}, y=2;\left(-\frac{1}{2},2\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}x-y=3\\ \frac{1}{2}x+y=3\end{array}\right. \\ x=4, y=1;\left(4,1\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}x-y=3\\ -3x+y=1\end{array}\right. \\ x=-2, y=-5;\left(-2,-5\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}\begin{array}{rcl}3x+3y+2z& =& 4\\ x-y-z& =& 0\end{array}\\ 2y-3z=-8\end{array}\right. \\ x=1, y=-1, z=2;\left(1,-1,2\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{r}\begin{array}{rcl}4x-z& =& 7\\ 8x+5y-z& =& 0\end{array}\\ -x-y+5z=6\end{array}\right. \\ x=2, y=-3, z=1;\left(2,-3,1\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}\begin{array}{rcl}3x+3y+2z& =& 4\\ x-3y+z& =& 10\end{array}\\ 5x-2y-3z=8\end{array}\right. \\ x=2, y=-2, z=2; \left(2,-2,2\right) \end{gathered}$$

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}\begin{array}{rcl}4x-5z& =& 6\\ 5y-z& =& -17\end{array}\\ -x-6y+5z=24\end{array}\right. \\ x=4, y=-3, z=2; \left(4,-3,2\right) \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{l}x+y=8\\ x-y=4\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{r}x+2y=-7\\ x+y=-3\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}5x-y=21\\ 2x+3y=-12\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{c}x+3y=5\\ 2x-3y=-8\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{c}3x=24\\ x+2y=0\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}4x+5y=-3\\ -2y=-8\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}3x-6y=2\\ 5x+4y=1\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{l}2x+4y=\frac{2}{3}\\ 3x-5y=-1\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}2x+y=1\\ 4x+2y=3\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}x-y=5\\ -3x+3y=2\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}2x-y=0\\ 4x+2y=12\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}3x+3y=-1\\ 4x+y=\frac{8}{3}\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}x+2y=4\\ 2x+4y=8\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}3x-y=7\\ 9x-3y=21\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}2x-3y=-1\\ 10x+y=11\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}3x-2y=0\\ 5x+10y=4\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}2x+3y=6\\ x-y=\frac{1}{2}\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}\frac{1}{2}x+y=-2\\ x-2y=8\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}\frac{1}{2}x+\frac{1}{3}y=3\\ \frac{1}{4}x-\frac{2}{3}y=-1\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}\frac{1}{3}x-\frac{3}{2}y=-5\\ \frac{3}{4}x+\frac{1}{3}y=11\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{c}3x-5y=3\\ 15x+5y=21\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}2x-y=-1\\ x+\frac{1}{2}y=\frac{3}{2}\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}\frac{1}{x}+\frac{1}{y}=8\\ \frac{3}{x}-\frac{5}{y}=0\end{array}\right.$

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{c}\frac{4}{x}-\frac{3}{y}=0\\ \frac{6}{x}+\frac{3}{2y}=2\end{array}\right.$

solve each system of equations. If the system has no solution, say that it is inconsistent.

$$\begin{gathered} x-y=6 \\ 2x-3z=16 \\ 2y+z=4 \end{gathered}$$

solve each system of equations. If the system has no solution, say that it is inconsistent. 

$$\begin{gathered} 2x+y=4 \\ -2y+4z=0 \\ 3x-2z=-11 \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent.  

$$\begin{gathered} x-2y+3z=7 \\ 2x+y+z=4 \\ -3x+2y-2z=-10 \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent.  

$$\begin{gathered} 2x+y-3z=0 \\ -2x+2y+z=-7 \\ 3x-4y-3z=7 \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$$\begin{gathered} x-y-z=1 \\ 2x+3y+z=2 \\ 3x+2y=0 \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$$\begin{gathered} 2x-3y-z=0 \\ -x+2y+z=5 \\ 3x-4y-z=1 \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent. 

$$\begin{gathered} x-y-z=1 \\ -x+2y-3z=-4 \\ 3x-2y-7z=0 \end{gathered}$$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$$\begin{gathered} 2x-3y-z=0 \\ 3x+2y+2z=2 \\ x+5y+3z=2 \end{gathered}$$