Precalculus, Chapter 7, 7.3, Section 7.3, Problem 26

You may use the substitution method to solve the system, hence, you need to use the first equation to write x in terms of y and z, such that:
x + y + z = 5 => x = 5 - y - z
You may now replace 5 - y - z for x in equation x - 2y + 4z = 13 , such that:
5 - y - z - 2y + 4z = 13 => -3y + 3z = 8
You may use the third equation, 3y + 4z = 13 , along with -3y + 3z = 8 equation, such that:
-3y + 3z + 3y + 4z = 8 + 13 => 7z = 21 => z = 3
You may replace 3 for z in equation 3y + 4z = 13:
3y + 12 = 13 => 3y = 1 => y = 1/3
You may replace 3 for z and 1/3 for y in equation x = 5 - y - z:
x = 5 -1/3 - 3 => x = 2 - 1/3 => x = 5/3
Hence, evaluating the solution to the given system, yields x = 5/3, y = 1/3, z = 3.