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Setting up equations


I'm trying to do things of the form
Solve[5 x + 6 y + 7 z == a x + b y + c z, {a, b, c}]

But since x,y,z are variables, what I really mean is
Solve[5==a && 6==b && 7==c], so I need to convert to this form

If I only have one variable, the following does what I need

LogicalExpand[a*x + b*x^2 + O[x]^3 == 2*x + 3*x^2 + O[x]^3]

But what to do if I have several variables?


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