Use Gauss’s Method to find a solution to the linear system:
$$\begin{align} 6x - 8 y &= 5 \\ 2 x + 4 y &= 0 \end{align}$$ For this problem we will walk you through one possible solution.
Note that if we add $\answer {2}$ times the second equation from the first we will eliminate the $y$ variable.
This gives us the single equation with only one unknown: $$\answer {10} x = \answer {5}$$ Solving for $x$ we have: $$x = \answer {0.5}$$ Back substitution to the original first equation gives: $$\answer {-8} y = \answer {2}$$ Solving for $y$ we have $$y = \answer {-0.25}$$

Use Gauss’s Method to find a solution to the linear system $$\begin{align} 2 u \phantom{+ 2 v} + w &= 7\\ u \phantom{+2 v} + w &= 4 \\ \phantom{2u +} 2 v + w &= 7 \end{align}$$

Use Gauss’s Method to determine how many solutions the system $$\begin{align} x + 2 y &= -3 \\ 2 x - 3 y &= 1 \end{align}$$ has.

Use Gauss’s Method to determine how many solutions the system $$\begin{align} \frac{1}{6} a + \frac{1}{2} b &= \frac{7}{6} \\ 2 a + 6 b &= 2 \end{align}$$ has.

Use Gauss’s Method to determine how many solutions the system $$\begin{align} x - 2 y &= -10 \\ \frac{1}{2} x - y &= - 5 \end{align}$$ has.

Use Gauss’s Method to determine how many solutions the system $$\begin{align} x_1 + x_2 - 2 x_3 \phantom{-2x_4} &= 3 \\ -x_1 -x_2 \phantom{-2x_3} - 2 x_4 &= -3 \end{align}$$ has.

Find the value such that there are many solutions to the system: $$\begin{align} x - 2y &= 4 \\ 3x - 6y &= \answer{12} \end{align}$$

Decide if the vector $\displaystyle \begin {pmatrix} 3 \\ 2 \end {pmatrix}$ is in the set $$\left \{ \begin {pmatrix}1\\0\end {pmatrix} + \begin {pmatrix} 1 \\ 1 \end {pmatrix} k \bigg | k\in \mathbb {R} \right \}$$

Decide if the vector $\displaystyle \begin {pmatrix} 0 \\ -4 \\ 2 \end {pmatrix}$ is in the set $$\left \{ \begin {pmatrix} 1\\1\\0 \end {pmatrix} x + \begin {pmatrix} 2 \\ 0 \\ 1 \end {pmatrix} y \bigg | x, y \in \mathbb {R} \right \}$$

There is a planet inhabited by watermeloners - they come in three colors red, green, and blue. There are 13 red, 15 blue, and 17 green. When two differently colored watermeloners meet they both change to the third color.

Can it happen that all of them assume the same color?