8.2 Linear equation systems

1. Question

Solve the equations system [math]\begin{cases}2x+3y-5z+40=0\\4x-8y+z-39=0\\-x-y-z+6=0\end{cases}[/math]

[math]x=[/math]
[math]y=[/math]
[math]z=[/math]

2. Question

Solve the equations system [math]\begin{cases}x+2y+3z=0\\4x+5y+6z=0\\7x+8y+9z=0\end{cases}[/math]

[math]x=[/math] [math]z+[/math]
[math]y=[/math] [math]z+[/math]
[math]z=[/math] all real numbers

3. Question

Solve the equations system [math]\begin{cases}4x+2y+z=0\\x+y+z=1\\9x+3y+z=-1\end{cases}[/math]

[math]x=[/math]
[math]y=[/math]
[math]z=[/math]

4. Question

Consider an augmented matrix [math]\begin{bmatrix}1&p&q&\Bigm|&5\\2q&2p&5&\Bigm|&15\\2p&7&3q&\Bigm|&6\end{bmatrix}[/math]. Suppose [math]x=1, y=-2, z=3[/math] is the solution to the augment matrix. Find [math]p[/math] and [math]q[/math]

[math]p=[/math]
[math]q=[/math]

5. Question

Solve the linear equation system [math]\begin{cases}x+y+z=6\\6x-y-2z=5\\-3x-2y+z=-5\end{cases}[/math] by RREF.

[math]x=[/math]
[math]y=[/math]
[math]z=[/math]

6. Question

(Calculator) Mr. Fang is traveling around Asia. Now he is taking an airplane home. On the airplane, he is sorting his bills:
[math]\bullet[/math] In Japan, he spent 2,000 on food, 4,000 on accommodation, and 500 on traffic every day.
[math]\bullet[/math] In Thailand, he spent 500 on food, 2,000 on accommodation, and 100 on traffic every day.
[math]\bullet[/math] In Singapore, he spent 1,500 on food, 3,000 on accommodation, and 400 on traffic every day.
He totally spent 43500 on food, 95000 on accommodation, and 10900 on traffic. How many days did he stay in Japan?