SOLID STATE PRESS
← Back to catalog
Solving Linear Equations cover
Coming soon
Coming soon to Amazon
This title is in our publishing queue.
Browse available titles
Mathematics

Solving Linear Equations

A High School & Early College Primer

Algebra shouldn't feel like a black box. But for a lot of students, linear equations hit a wall somewhere between one-step problems and the moment fractions show up — or when both sides of the equation have variables and there's no obvious next move. If you have a test coming up, a homework set that isn't clicking, or you're helping a student who's stuck, this guide cuts straight to what matters.

TLDR: Solving Linear Equations walks through every stage of solving linear equations in one variable, in the order a student actually needs them. It starts with the core idea of keeping an equation balanced, builds through one-step and two-step problems, then tackles distribution, combining like terms, and the strategies that make solving equations with fractions or decimals far less painful. It covers how to handle variables on both sides of an equation, and it explains the two special cases most students find confusing — equations with no solution and equations with infinitely many solutions. The final section shows how to translate word problems into equations and solve them using the same methods.

This is a focused, 15-page primer. No filler chapters, no review of topics you already know. Every concept is defined plainly, every method is shown with worked examples, and common mistakes are flagged where they happen — not buried in footnotes.

If you want a quick algebra review before a math test or a clear foundation before moving into systems of equations, pick this up and work through it in an afternoon.

What you'll learn
  • Identify what makes an equation linear and what 'solving' it means
  • Use inverse operations to isolate a variable in one-step and multi-step equations
  • Solve equations with parentheses, fractions, decimals, and variables on both sides
  • Recognize and interpret equations with no solution or infinitely many solutions
  • Translate word problems into linear equations and check solutions reliably
What's inside
  1. 1. What Is a Linear Equation?
    Defines linear equations in one variable, what a solution is, and the core idea of keeping an equation balanced.
  2. 2. One-Step and Two-Step Equations
    Walks through isolating a variable using inverse operations, starting from the simplest cases.
  3. 3. Multi-Step Equations: Distribution and Combining Like Terms
    Tackles equations that need simplification before isolation, including parentheses and like terms on the same side.
  4. 4. Variables on Both Sides, Fractions, and Decimals
    Strategies for moving variables across the equals sign and clearing fractions or decimals to make work cleaner.
  5. 5. Special Cases: No Solution and Infinitely Many Solutions
    How to recognize identities and contradictions, and what they mean about the equation.
  6. 6. From Words to Equations: Setting Up and Solving Problems
    Translates common word-problem patterns into linear equations and applies the solving methods from earlier sections.
Published by Solid State Press
Solving Linear Equations cover
TLDR STUDY GUIDES

Solving Linear Equations

A High School & Early College Primer
Solid State Press

Solving Linear Equations

A High School & Early College Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you're sitting in Algebra 1 or Pre-Algebra and need a clear, direct algebra 1 equations study guide for teens that cuts straight to what matters, this book is for you. Same if you're a college student in a remedial or intro math course, or a parent who wants to sit down and actually help with tonight's homework.

This book covers how to solve linear equations step by step — one-step and two-step equations, multi-step problems using distribution and combining like terms, variables on both sides of equation problems, solving equations with fractions and decimals in high school math, and the tricky special cases of linear equations with no solution or infinite solutions. It also walks through turning word problems to linear equations and solving them. About 15 pages. No filler.

Read it straight through, work every example alongside the text, then hit the practice problems at the end. Think of it as a quick algebra review before a math test — compact, focused, and done.

Contents

  1. 1 What Is a Linear Equation?
  2. 2 One-Step and Two-Step Equations
  3. 3 Multi-Step Equations: Distribution and Combining Like Terms
  4. 4 Variables on Both Sides, Fractions, and Decimals
  5. 5 Special Cases: No Solution and Infinitely Many Solutions
  6. 6 From Words to Equations: Setting Up and Solving Problems
Chapter 1

What Is a Linear Equation?

A linear equation in one variable is a statement that two expressions are equal, where the variable appears only to the first power — no exponents, no square roots, no $x^2$ terms. The word linear comes from the fact that these equations describe straight lines when graphed, but for now the algebra is what matters.

Here is a typical example: $3x + 5 = 17$. It has three ingredients worth naming precisely.

  • A variable is a letter that represents an unknown number — here, $x$.
  • A coefficient is the number multiplying the variable — here, $3$ (the $3$ in $3x$).
  • A constant is a fixed number with no variable attached — here, $5$ on the left and $17$ on the right.

An equation is linear in one variable as long as every term is either a constant or a coefficient times the variable to the first power. Something like $x^2 + 3 = 7$ is not linear — the squared exponent disqualifies it.

What "Solving" Means

Solving an equation means finding every value of the variable that makes the equation a true statement. That value is called a solution (also called a root). For $3x + 5 = 17$, the solution is $x = 4$, because substituting $4$ in gives $3(4) + 5 = 12 + 5 = 17$ — a true statement.

Checking by substitution is not optional busywork. It is the only reliable way to confirm your answer is correct, especially when the arithmetic gets complicated later. Make a habit of it now.

A common misconception is that the goal is to "find $x$" in some abstract sense — as if $x$ is always hiding on the left side. Actually, a solution is just a number that satisfies the equation, meaning both sides become equal when you plug it in. The variable's position does not matter; $4 = x$ and $x = 4$ say the same thing.

The Balance Principle

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