#### Chapter 4 System of Linear Equations

**Section 4.5 Final Test**

# 4.5.1 Final Test Module 4

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##### **Exercise 4.5.1 **

Find the solution set of the following system of linear equations:

$\begin{array}{ccc}\multicolumn{1}{c}{-x+2y}& =\hfill & -5\hspace{0.5em},\hfill \\ \multicolumn{1}{c}{3x+y}& =\hfill & 1\hspace{0.5em}.\hfill \end{array}$

The solution set

The solution set

is empty, | |

contains exactly one element: $x=$ , $y=$ , | |

contains an infinite number of solution pairs $(x;y)$. |

##### **Exercise 4.5.2 **

Find the two-digit number such that its digit sum is 6 and exchanging the tens and the units digit results in a number which is 18 less. Answer:

.

.

##### **Exercise 4.5.3 **

Find the value of the real parameter $\alpha $ for which the following system of linear equations

$\begin{array}{ccc}\multicolumn{1}{c}{2x+y}& =\hfill & 3\hspace{0.5em},\hfill \\ \multicolumn{1}{c}{4x+2y}& =\hfill & \alpha \hfill \end{array}$

has an infinite number of solutions.

Answer: $\alpha =$

.

has an infinite number of solutions.

Answer: $\alpha =$

.

##### **Exercise 4.5.4 **

The following figure shows two lines in two-dimensional space. Find the two equations describing the lines.

Line 1: $y=$

,

Line 2: $y=$ .

What is the number of solutions of the corresponding system of linear equations?

The system of linear equations has

Line 1: $y=$

,

Line 2: $y=$ .

What is the number of solutions of the corresponding system of linear equations?

The system of linear equations has

no solution, | |

exactly one solution, or | |

an infinite number of solutions. |

##### **Exercise 4.5.5 **

Find the solution set of the following system of linear equations consisting of three equations in three variables:

$\begin{array}{ccc}\multicolumn{1}{c}{x+2z}& =\hfill & 3\hspace{0.5em},\hfill \\ \multicolumn{1}{c}{-x+y+z}& =\hfill & 1\hspace{0.5em},\hfill \\ \multicolumn{1}{c}{2y+3z}& =\hfill & 5\hspace{0.5em}.\hfill \end{array}$

The solution set

The solution set

is empty, | |

contains exactly one solution: $x=$ , $y=$ , $z=$ , | |

contains an infinite number of solutions $(x;y;z)$. |

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