Question

Find the solution to this system of equations x+y=1
2x-y+z=1
x+2y+z=8/3

Answer 1

`\left\{\begin{array}{ccc}x+y=1&\to x=1-y\\2x-y+z=1\\x+2y+z=(8)/(3)\end{array}\right\\\text{substitute x=1-y to the other equations}\\ \left\{\begin{array}{ccc}2(1-y)-y+z=1\\1-y+2y+z=(8)/(3)\end{array}\right\\ \left\{\begin{array}{ccc}2-2y-y+z=1&|-2\\1+y+z=(8)/(3)&|-1\end{array}\right\\ \left\{\begin{array}{ccc}-3y+z=-1\\y+z=(5)/(3)\end{array}\right\\\text{substract both sides of the equations}\\-4y=-(8)/(3)\ \ \ \ |:(-4)\\y=(2)/(3)`


`\text{substitute the value of y to the second equation}\\\\(2)/(3)+z=(5)/(3)\ \ \ \ |-(2)/(3)\\\\z=(3)/(3)\to z=1\\\\\text{substitute the value of y to the equation}\ x=1-y\\\\x=1-(2)/(3)\to x=(1)/(3)\\\\\text{Answer:}\ \left\{\begin{array}{ccc}x=(1)/(3)\\\\y=(2)/(3)\\\\z=1\end{array}\right`