Two-Step Equations in Context

A two-step equation is straightforward once it is written down. The real skill is producing it from a sentence — turning "multiply a number by 2 and add 5" into symbols you can actually work with.

Concept Translating a sentence

"What number, multiplied by 2 and then increased by 5, gives 9?"

Let the unknown be

x

"multiplied by 2" becomes

2x

"increased by 5" becomes

2x + 5

"gives 9" becomes

= 9

⟹ 2x + 5 = 9

Work through the sentence in order, converting one phrase at a time. Nothing is left to guesswork.

Example Solving 2x + 5 = 9

Subtract 5 from both sides:

2x = 4

Divide both sides by 2:

x = 2

Check:

2 \times 2 + 5 = 4 + 5 = 9

⟹ x = 2

The constant goes first, then the coefficient — the same order every time.

Example When the constant is negative

Solve $3x - 5 = 10$ .

The 5 is subtracted, so add 5 to both sides:

3x = 15

Divide both sides by 3:

x = 5

Check:

3 \times 5 - 5 = 15 - 5 = 10

⟹ x = 5

Whatever sign the constant carries, you remove it with the opposite operation.

Example A shopping budget

Someone has 65 riyals. A bag costs 23 riyals and each book costs 14. If they buy the bag, how many books can they afford?

Let

x

be the number of books.

The books cost

14x

, the bag costs 23.

Everything must total 65:

14x + 23 = 65

Subtract 23 from both sides:

14x = 42

Divide both sides by 14:

x = 3

Check:

14 \times 3 + 23 = 42 + 23 = 65

⟹ the bag and 3 books

Notice that the fixed cost and the repeating cost play different roles: 23 appears once, while 14 is multiplied by the unknown. Identifying which is which is what makes the equation correct.

Concept The method in short

1. Name the unknown.

2. Translate the sentence one phrase at a time.

3. Clear the constant from both sides.

4. Divide both sides by the coefficient.

5. Substitute back to check.

Note Mistakes to avoid

Multiplying the fixed cost by the unknown as well as the repeating one.

Dividing before the constant has been cleared.

Subtracting a constant that was already negative instead of adding it.

Translating the phrases out of order.

Reporting a number without saying what it counts.

Summary

Translate a sentence phrase by phrase into an equation.
Clear the constant first, then divide by the coefficient.
A subtracted constant is removed by adding it to both sides.
In budget problems, the fixed cost stands alone and the repeating cost multiplies the unknown.
Always substitute the answer back into the original equation.