Justifying a Step, and Writing a Reason That Earns Credit

In this lesson you will: Write a reason that names a rule instead of a goal; Justify replacing one equation of a system by a combination of the two; Justify a conclusion in context, including the direction you rounded.

A reason has three parts

A justification says what you did, what permits it, and what it produced. Drop any one of the three and the sentence stops being checkable by a reader.

The most common failure is not laziness. It is stating the goal instead of the reason — \"to get rid of the 4\" describes what you wanted, not why the move was allowed.

Why you may add a multiple of one equation to another

Elimination looks like a trick until you can say why it is legal. Take 2x+3y=12 and x-y=1. Replace the first equation by itself minus twice the second: 2x+3y-2(x-y)=12-2(1), which is 5y=10, so y=2 and then x=3.

The justification has two halves. First, any pair (x,y) satisfying both original equations satisfies the combination, because equals subtracted from equals leave equals. Second, the move is reversible — add 2 times the second equation back and you recover the first — so no solutions were invented either. Same solution set, easier system. That is A.REI.5 in one sentence.

Justifying a conclusion, not just a step

Some questions do not ask you to defend a line of algebra; they ask you to defend an answer about a situation. Those justifications have to mention the situation.

Two gym plans: Plan A charges $30 to join and $22 a month, so A(m)=30+22m; Plan B charges $25 a month with nothing to join, so B(m)=25m. They cost the same when 30+22m=25m, that is 30=3m, so m=10 months at $250.

\"Plan A is better\" is not a justified conclusion. \"Plan A costs less for any stay longer than 10 months, because the plans are equal at 10 and A gains only $22 a month against B's $25\" is one.

Common mistake — A reason that restates the step

Asked to justify going from 3x=15 to x=5, a student writes \"I divided by 3 because I needed to divide by 3.\"

It feels like an answer because it is a true sentence about the work, and it uses the vocabulary of the step. Under time pressure, saying what you did in slightly different words feels like explaining it.

A reason has to come from outside the step. The step is \"divided both sides by 3\"; the reason is \"the division property of equality permits dividing both sides by the same nonzero number without changing the solution set.\"

The same trap wears a second costume: \"because it works,\" offered after checking a single value. Checking one number confirms that number. It says nothing about whether the step was legal or whether other solutions exist.

How to catch it: Cover the step with your thumb and read the reason on its own. If it no longer tells a stranger anything they could act on, it is a restatement. A usable reason names a property, cites a definition, or points at a condition.