Global Admission Prep
GMAT · AlgebraEasyEasy Problem Solving · ~2 min

Solving a System of Two Linear Equations from a Word Problem

Answer

The answer is C — Lucia bought 7 notebooks. Let n = notebooks and p = pens. Then n + p = 12 and 4n + 2p = 38. Substituting p = 12 − n gives 2n + 24 = 38, so n = 7.

Problem

A campus store sells notebooks for $4 each and pens for $2 each. Lucia buys a total of 12 items and spends exactly $38. How many notebooks did she buy?

  • A5
  • B6
  • C7✓ Correct
  • D8
  • E9

Step-by-step solution

  1. Define variables. Let n = the number of notebooks and p = the number of pens.

  2. Write two equations. Total items: n + p = 12. Total cost: 4n + 2p = 38.

  3. Substitute. From the first equation, p = 12 − n. Plug into the cost equation: 4n + 2(12 − n) = 38.

  4. Solve. 4n + 24 − 2n = 38, so 2n = 14 and n = 7. Lucia bought 7 notebooks and 5 pens. The answer is C.

Key takeaway: Two unknowns need two equations. Translate the 'total count' and the 'total value' as separate equations, then substitute — it is faster and safer than testing answer choices.

Frequently asked

How do I solve two-variable word problems quickly on the GMAT?

Assign a variable to each unknown, write one equation for the total count and one for the total value, then substitute the isolated variable. Most GMAT systems solve in two or three lines.

Should I use substitution or elimination here?

Use substitution when one variable is easy to isolate, as in this problem. Use elimination when the coefficients can be made to cancel directly. Both methods give n = 7.

Is this a Problem Solving or a Data Sufficiency question?

This is a Problem Solving question because it asks for a specific numerical value. A Data Sufficiency version would instead ask whether the two statements are enough to determine the number of notebooks.