1) Let's carefully read the statements so that we write accurate equations.
"The sum of three numbers is 17"
x+y+z=17
"Two times the smallest is 4 less than the largest"
2z=x-4
" the sum of the largest and smallest is 13"
x+z=13
Note that we have stated the largest one to be "x" and the smallest one to be "z"
2) Setting this Linear system we have:
[tex]\begin{gathered} \mleft\{\begin{matrix}x+y+z=17 \\ 2z=x-4 \\ x+z=13\end{matrix}\mright. \\ \end{gathered}[/tex]Now, let's solve it by performing some algebraic manipulations:
[tex]\begin{gathered} \{\begin{matrix}x+y+z=17 \\ 2z-x-4 \\ x+z=13\end{matrix} \\ -x+2z=-4 \\ x+z=13 \\ -------- \\ 3z=9 \\ \frac{3z}{3}=\frac{9}{3} \\ z=3 \\ x+3=13 \\ x+3-3=13-3 \\ x=10 \\ x+y+z=17 \\ 10+y+3=17 \\ 13+y=17 \\ 13+y-13=17-13 \\ y=4 \end{gathered}[/tex]Note that we started solving two equations and then plugging the result into original equations, we ended up finding the variables.
Thus the answer is:
[tex]x=10,y=4,z=3[/tex]