How do I check whether an equation matches a word problem?

Start by naming the quantity you are trying to find. Then restate the important information in a short sentence and compare that sentence with your equation. OpenStax recommends understanding the prompt, naming the unknown, restating the information, translating the relationship into an equation, and interpreting the result [S1].

For each part of your equation, ask: What does this number stand for? What does this variable stand for? What relationship does the operation express? If you cannot point to a specific quantity or relationship in the prompt, pause and reconsider that part.

For example, suppose a problem says that 18 is one-half of the original amount. Let p be the original amount. The equation 18 = 1/2 p matches the stated relationship: 18 equals half the original amount [S1]. Reading the equation aloud—“18 is one-half of p”—makes the match easier to inspect.

Translate the relationship, not just an isolated word. First identify what each quantity means; then describe how they are related in a complete sentence. For instance, “the total is 5 more than twice the number” can be represented with T = 2n + 5, if T is the total and n is the number. Read it back: “T is 5 more than twice n.” If that does not say what the prompt says, revise it.

Watch the order of quantities, especially in comparisons. “A is 5 more than B” means A = B + 5; the equation should still make sense when you read it aloud. Keep units attached in your notes, too: dollars, people, or minutes can help reveal when you have added unlike quantities or assigned a variable the wrong meaning. These checks are reasoning aids, not a guarantee that an equation is correct.

Substitution checks whether your proposed value makes your equation true: put the value in, simplify both sides, and see whether they are equal [S2]. That is useful, but it checks the solution against the equation—not, by itself, whether the equation captured the word problem correctly [S2].

So do two checks. First, verify the algebra by substitution. Second, return to the prompt: does the result answer the quantity asked for, use the right units, and fit the stated relationship? A useful extra check is to try a simple hypothetical value in the relationship you wrote and see whether the equation behaves as your sentence says it should. This is a reasoning technique, not a test guaranteed to catch every modeling error; the cited material does not establish a general simple-case test [S1].

Assume a problem says, “A club has 12 members after 4 more join. How many members were there before they joined?” Let x be the number before the new members arrived. The relationship is “starting members plus 4 equals 12,” so write x + 4 = 12. Solve to get x = 8. Check the equation: 8 + 4 = 12, which is true. Then check the story: 8 members before, followed by 4 joining, gives 12; the answer is in members and answers the question.

Reusable model check:

1. Name the unknown and state what its units are.
2. Mark the given quantities and the relationship between them.
3. Write what the equation says in ordinary words; compare it with the prompt.
4. Check that every number and operation has a reason in the prompt.
5. Substitute the solution into the equation, then interpret it in the story.

The substitution check follows the equation-verification method described by OpenStax [S2]. Keep the modeling check separate: a calculation can be internally correct even when the starting equation does not represent the prompt.