Puzzle

# set up triangle
var rows = 5
var tri = rows.of {|i| (i+1).of { Hash(x => 0, z => 0, v => 0, rhs => nil) } }
tri[0][0]{:rhs} = 151
tri[2][0]{:rhs} = 40
tri[4][0]{:x} = 1
tri[4][1]{:v} = 11
tri[4][2]{:x} = 1
tri[4][2]{:z} = 1
tri[4][3]{:v} = 4
tri[4][4]{:z} = 1
 
# aggregate from bottom to top
for row in (tri.len ^.. 1) {
    for col in (^tri[row-1]) {
        [:x, :z, :v].each { |key|
            tri[row-1][col]{key} = (tri[row][col]{key} + tri[row][col+1]{key})
        }
    }
}
 
# find equations
var eqn = gather {
    for r in tri {
        for c in r {
            take([c{:x}, c{:z}, c{:rhs} - c{:v}]) if defined(c{:rhs})
        }
    }
}
 
# print equations
say "Equations:"
say " x +  z = y"
for x,z,y in eqn { say "#{x}x + #{z}z = #{y}" }
 
# solve
var f = (eqn[0][1] / eqn[1][1])
{|i| eqn[0][i] -= (f * eqn[1][i]) } << ^3
f = (eqn[1][0] / eqn[0][0])
{|i| eqn[1][i] -= (f * eqn[0][i]) } << ^3
 
# print solution
say "Solution:"
var x = (eqn[0][2] / eqn[0][0])
var z = (eqn[1][2] / eqn[1][1])
var y = (x + z)
say "x=#{x}, y=#{y}, z=#{z}"

Output:

Equations:
 x +  z = y
7x + 7z = 91
2x + 1z = 18
Solution:
x=5, y=13, z=8