Linear Equations Word Problems

A High School & College Primer on Translating Words into Math

Word problems are where most algebra students hit a wall. The math itself is manageable — it's the translation from English sentences into equations that trips people up. If you've ever stared at a problem about two trains leaving stations, a jar of mixed coins, or someone's age in ten years and had no idea where to start, this guide is for you.

**TLDR: Linear Equations Word Problems** is a focused, 10–20 page primer that covers every major category you'll encounter in a high school or early college algebra course: number and age problems, coin and percent problems, distance-rate-time problems, and mixture problems using systems of two equations. Each section builds a clear translation toolkit — turning phrases like "five more than twice a number" into expressions you can actually work with — and walks through fully solved examples before asking you to practice.

This guide is written for students in grades 9–12 and college freshmen who need to get up to speed fast, as well as parents and tutors looking for a clean, no-fluff reference to use in a single study session. It covers the step-by-step process for how to solve word problems with equations, the most common mistakes students make (and exactly why they happen), and a reusable checklist you can apply to any problem on a quiz or standardized test.

No padding. No review of things you already know. Just the concepts, the method, and the worked examples — so you can walk into your next exam with a real plan.

What you'll learn

Translate everyday language into linear expressions and equations using a clear variable definition
Set up and solve word problems involving age, money, mixtures, distance-rate-time, and consecutive integers
Use systems of two linear equations to handle problems with two unknowns
Check answers against the original wording to catch common setup mistakes

What's inside

1. From Words to Equations: The Translation Toolkit

How to read a word problem, define variables, and convert phrases into linear expressions and equations.
2. Number, Age, and Consecutive Integer Problems

Setting up one-variable linear equations for the classic 'find the number' and age-relationship problems.
3. Money, Coins, and Percent Problems

Using linear equations for problems about prices, coins, discounts, tax, and simple interest.
4. Distance, Rate, and Time Problems

Applying d = rt to problems with travelers moving toward, away from, or alongside each other.
5. Mixture Problems and Systems of Two Equations

Handling two-unknown problems like mixing solutions, blending prices, and ticket sales using systems of linear equations.
6. Strategy, Common Mistakes, and Why It Matters

A reusable problem-solving checklist, the most frequent student errors, and where these skills show up later.

Published by Solid State Press

TLDR STUDY GUIDES

Linear Equations Word Problems

A High School & College Primer on Translating Words into Math

Solid State Press

Linear Equations Word Problems

A High School & College Primer on Translating Words into Math

Published by Solid State Press.

Part of the TLDR Study Guides series.

Who This Book Is For

If you are a high school student who freezes the moment a math problem becomes a paragraph, this is your book. It is written for anyone taking Algebra I, Algebra II, or a college developmental math course — and for students prepping for the SAT, ACT, or a state standardized exam where word problems reliably show up and reliably cost points.

This algebra word problems high school study guide covers every major problem type: how to translate word problems into equations, consecutive integer problems, age problems, coin and percent problems, distance rate and time problems, and mixture problems using systems of equations. Each section works through the logic step by step, with fully solved examples and plain-language explanations. About 15 pages, zero filler.

Start at the beginning and read straight through — the sections build on each other. Work every example yourself before reading the solution. Then hit the problem set at the end to find out what actually stuck.

1 From Words to Equations: The Translation Toolkit
2 Number, Age, and Consecutive Integer Problems
3 Money, Coins, and Percent Problems
4 Distance, Rate, and Time Problems
5 Mixture Problems and Systems of Two Equations
6 Strategy, Common Mistakes, and Why It Matters

Chapter 1

From Words to Equations: The Translation Toolkit

Every word problem is a puzzle in two layers: first an English puzzle, then a math puzzle. Most students who struggle with word problems get stuck on the first layer — they try to do algebra before they've finished reading carefully. The fix is a repeatable translation process, and that's what this section gives you.

Step 1: Read the whole problem before writing anything

Read once for the big picture. Ask: What is this problem about? What is the unknown? What relationships are stated? Only on a second pass should you start writing math. Rushing to an equation before you understand the situation is the single most common reason students set up problems incorrectly.

Step 2: Define your variable with a "let statement"

A variable is a letter that stands in for an unknown quantity. Before writing any expression or equation, write a let statement — a plain English sentence that says exactly what your variable represents, including units.

Good: Let $x$ = the number of miles driven. Bad: Let $x$ = miles.

The difference matters. A sloppy let statement causes errors when a problem has multiple quantities (age now vs. age in five years, for example). Being precise costs two seconds and saves five minutes of backtracking.

Step 3: Translate phrases into expressions

A linear expression is a combination of constants and variables connected by addition or subtraction, where no variable is multiplied by another variable or raised to a power. Examples: $3x + 7$, $2n - 1$, $x$.

Certain English phrases map reliably onto arithmetic operations. Learn these patterns:

English phrase	Mathematical meaning
"more than," "increased by," "sum of"	addition ($+$)
"less than," "decreased by," "difference"	subtraction ($-$)
"times," "product of," "twice," "triple"	multiplication ($\times$)
"divided by," "quotient of," "per"	division ($\div$)
"is," "equals," "is the same as"	equals sign ($=$)

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon