#### 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:

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

The solution set
 is empty, contains exactly one element: $x=$ , $y=$ , contains an infinite number of solution pairs $\left(x;y\right)$.

##### 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

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

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
 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:

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

The solution set
 is empty, contains exactly one solution: $x=$ , $y=$ , $z=$ , contains an infinite number of solutions $\left(x;y;z\right)$.

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