# Solve LinearProblem

There some LinearProblem example.

## Simple linear problem

solve function:

$$$\left\{\begin{matrix} x+y+z = 3\\ x + 4y + 9z = 14 \\ x + 2y + 3z = 6\end{matrix}\right.$$$

julia code:

@variables x,y,z
eqs = [
x + y + z ~ 3,
x + 4y + 9z ~ 14,
x + 2y + 3z ~ 6
]
vars = Dict(x => 1.0, y => 0.0, z => 0.0)
pro = LinearProblem(eqs,vars)
res = solve(pro)

res = Dict{Num, Float64}(
y => 0.9999999999999996,
z => 1.0000000000000002,
x => 1.0000000000000002
) 

## Using NLProblem to solve LP

Actually, the way EquationsSolver using to solve NLP is Newton iterative method. It also can solve LP, because LP can be treated as a special NLP.

We can make a problem like this. The value of all symbols is 1.0.

N = 100
@variables x[N]
eqs = Vector{Equation}([])
for i in 1:N
rand_num = rand(1:20, Vector{Int}, 3)
s = sum(rand_num)
if i == 1
push!(eqs, sum([rand_num[j-i+1] * x[j] for j in i:i+2]) ~ s)
elseif i == N
push!(eqs, sum([rand_num[j-i+3] * x[j] for j in i-2:i]) ~ s)
else
push!(eqs, sum([rand_num[j-i+2] * x[j] for j in i-1:i+1]) ~ s)
end
end
vars=Dict([x[i]=>0.0 for i in 1:N])
pro = NLProblem(eqs,vars)
res = solve(pro)