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Unknown number related questions in linear equations

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Unknown-Number Word Problems

An unknown-number word problem, such as finding consecutive integers that sum to a given total, is solved by naming the first unknown and writing the others in terms of it. See a worked example with three consecutive integers, solved step by step as a linear equation.

Setting up an unknown-number word problem

An unknown-number word problem describes a relationship between numbers — often consecutive integers — and asks you to find them. As with any word problem, the key step is naming the first unknown so every other quantity can be written in terms of it.

Worked example

Three consecutive integers add up to 72. Let n be the first integer; the next two are then n + 1 and n + 2, since consecutive integers increase by 1 each time.

Setting up an unknown-number word problem Three consecutive integers add up to 72. If n is the first integer, the next two are n plus 1 and n plus 2. The equation is n plus n plus 1 plus n plus 2 equals 72, which simplifies to 3n plus 3 equals 72, so n equals 23. The problem Three consecutive integers add up to 72. Let n = the first integer, so the others are n+1 and n+2. Setting up the equation n + (n + 1) + (n + 2) = 72 3n + 3 = 72 3n = 69 → n = 23
Naming the first unknown lets every other integer be written in terms of it.

Adding all three: n + (n + 1) + (n + 2) = 72, which simplifies to 3n + 3 = 72, so 3n = 69 and n = 23. The three integers are 23, 24, and 25.

Variations

The same setup handles consecutive even or odd integers — you just increase each term by 2 instead of 1. This naming-then-equation approach is the same one used for money word problems, distance-and-time word problems, and rectangle word problems.

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