Fill in the blank with the correct particle to complete the phrasal verb and convey the meaning of support: Do your best and we will back you ______.

Introduction / Context:
This question tests your understanding of phrasal verbs, which are very common in everyday English. The phrase "back you ______" must be completed with the correct particle so that the expression accurately conveys support or encouragement. Such questions are frequently used in competitive exams to check whether candidates are comfortable with natural idiomatic usage rather than only literal meanings.

Given Data / Assumptions:

• The sentence is "Do your best and we will back you ______."
• The options are "on", "in", "out", and "up".
• The intended meaning is that someone will support you or stand behind you if you try your best.
• We must identify which complete phrasal verb is standard and idiomatic.

Concept / Approach:
Phrasal verbs consist of a main verb plus one or more particles (prepositions or adverbs) that often change the meaning of the base verb. The verb "back" combines with different particles to create different meanings. We need the combination that means "support someone", which is a fixed and familiar expression in modern English.

Step-by-Step Solution:
Step 1: Recall common phrasal verbs with "back". One of the most frequent is "back up", which means "support", "confirm", or "reinforce".Step 2: Insert "up" into the blank to get "Do your best and we will back you up". This fits the intended meaning perfectly: "we will support you".Step 3: Test "back you on". This combination is not a standard phrasal verb in this context and sounds unnatural.Step 4: Test "back you in". While we might say "back you in a project" in a longer phrase, the usual idiomatic pattern in a short sentence is still "back you up".Step 5: Test "back you out". This would suggest removing someone from a situation, which clearly does not match the context of support.Step 6: Conclude that "up" is the correct particle, so option D is right.

Verification / Alternative check:
We can verify by examining natural usage patterns. People commonly say "I will back you up", "My friends backed me up", or "The evidence backs up this claim". These expressions all use "back up" in the sense of supporting someone or something. Other particles like "on", "in", and "out" do not produce widely accepted phrasal verbs with this supportive meaning in such a simple sentence. Therefore, the answer "up" is secure.

Why Other Options Are Wrong:
Option A, "on", does not form a typical phrasal verb "back on" meaning "support" in this short structure, so "back you on" is incorrect on its own.
Option B, "in", might appear in extended expressions like "back you in this venture", but the fixed and expected phrase in neutral English is still "back you up", especially when the rest of the sentence already implies general support.
Option C, "out", would lead to "back you out", which suggests withdrawing you from something rather than supporting you; it is not the intended idea.

Common Pitfalls:
Students often try to choose particles based only on literal meaning, thinking that "in" or "on" could somehow fit the idea of standing with someone. However, phrasal verbs are largely idiomatic, and the only reliable method is to remember common combinations. Another pitfall is to assume that if a combination is occasionally heard in some context, it must always be correct, but exam questions usually focus on the most accepted and widely used form.

Final Answer:
The correct phrasal verb is "back you up", so the answer is option D, "up".