1. ## Sysyem

Solve the system in R :
x + y = 1
y + z = 1
x + z = 1

2. Guideline

express y as function of x
put it in second equation
express z as function of x
put in third equation
find z
put z in second equation
find y
put y in first equation
find x

3. Hello, dhiab!

Solve the system in R:

. . $\displaystyle \begin{array}{cccccccc} x & + & y & & & = & 1 & [1] \\ & & y & + & z & = & 1 & [2] \\ x & & & + &z & = & 1 & [3] \end{array}$

$\displaystyle \begin{array}{ccccc}\text{Subtract [1]-[2]:} & x - z &=& 0 \\ \text{Add [3]:} & x + z &=& 1 \end{array}$

And we have: .$\displaystyle 2x \:=\:1 \quad\Rightarrow\quad \boxed{x \:=\:\tfrac{1}{2}}$

Substitute into [1]: .$\displaystyle \tfrac{1}{2} + y \:=\:1 \quad\Rightarrow\quad\boxed{ y \:=\:\tfrac{1}{2}}$

Substitute into [2]: .$\displaystyle \tfrac{1}{2} + z \:=\:1 \quad\Rightarrow\quad \boxed{z \:=\:\tfrac{1}{2}}$

4. Another approach:

Add all the equations: $\displaystyle 2(x+y+z)=3\Rightarrow x+y+z=\frac{3}{2}$.

$\displaystyle x+y=1\Rightarrow z=\frac{1}{2}$

$\displaystyle y+z=1\Rightarrow x=\frac{1}{2}$

$\displaystyle x+z=1\Rightarrow y=\frac{1}{2}$