8.2 Real-World Linear Equation Word Problems & Writing Equations from Context

Why Real-World Equation Word Problems Matter on the GED

Assessment target A.2.b and A.2.c ask you to write a linear equation from a real-world context and then solve it — and the GED Testing Service's own Assessment Guide is explicit that these items are drawn from workplace and consumer contexts (cell phone bills, wages, rental fees, shopping) as well as everyday number relationships (consecutive integers, comparisons between quantities). This is not a side skill: on the actual test, the majority of A.2 items are presented as a paragraph of text rather than a bare equation, so translating words into algebra correctly is often the harder half of the problem — the solving step itself is usually a one- or two-step equation once it's set up. Missing the translation means missing the question even if your arithmetic is flawless.

Translating Words Into Algebra

Every word problem is built from a small vocabulary of phrases that map directly onto math symbols. Memorize this translation table:

The most common translation error is reversing subtraction order. "5 less than a number" means you start with the number and subtract 5, so it translates to x − 5, not 5 − x. Reading the phrase in the order it's written, rather than translating word-by-word left to right, avoids this trap.

A Repeatable Process for Word Problems

1. Read the whole problem first, then identify exactly what quantity is unknown and assign it a variable (state what the variable represents in a short note, e.g., "let t = number of texts").
2. Separate constants from rates. A constant is a one-time or fixed amount (a monthly fee, a flat deposit); a rate is multiplied by the unknown (a per-unit cost, a per-mile charge).
3. Translate the sentence into an equation, term by term, using the table above.
4. Solve using the standard multi-step order from Section 8.1: distribute, clear fractions, combine like terms, isolate the variable.
5. Check your answer in context — does the number make sense (a whole number of texts, a positive number of miles), and does it satisfy the original sentence?

Worked Example 1 — Fixed Fee Plus Rate (Consumer Context)

"A cell phone plan charges a $25 monthly fee plus $0.10 per text message. If Maria's bill was $34, how many texts did she send?"

• Let t = number of texts sent.
• Fixed fee ($25) plus rate ($0.10 per text) equals total: 25 + 0.10t = 34
• Subtract 25: 0.10t = 9
• Divide by 0.10: t = 90 texts

Worked Example 2 — Consecutive Integers

"The sum of three consecutive integers is 51. Find the integers."

• Let n = the first integer, so the next two are (n + 1) and (n + 2).
• n + (n + 1) + (n + 2) = 51
• Combine like terms: 3n + 3 = 51
• Subtract 3: 3n = 48
• Divide by 3: n = 16, so the integers are 16, 17, and 18.

Worked Example 3 — Direct Phrase Translation

"5 less than twice a number is 13. Find the number."

• Twice a number is 2x. "5 less than" that quantity is 2x − 5.
• Equation: 2x − 5 = 13
• Add 5: 2x = 18
• Divide by 2: x = 9

Common Problem Templates on the GED

Recognizing which template a paragraph fits is often faster than re-deriving the equation from scratch every time — practice spotting the pattern within the first sentence or two of the problem.

Worked Example 4 — Comparison Problems

"Ana worked 3 more hours than twice the hours Sam worked. Together they worked 33 hours. How many hours did Sam work?"

• Let s = Sam's hours. Ana's hours are described in terms of Sam's: 2s + 3.
• Together they worked 33 hours, so: s + (2s + 3) = 33
• Combine like terms: 3s + 3 = 33
• Subtract 3: 3s = 30
• Divide by 3: s = 10 hours (and Ana worked 2(10) + 3 = 23 hours)

Comparison problems like this one are a favorite GED format because they force you to translate two related quantities with only one variable — define the simpler quantity first, then build the second quantity's expression from it before writing the final equation.