Key Concepts in Solving Linear Equations

Why This Matters

Linear equations are the foundation of algebra—and they show up everywhere on your exam. You're being tested on your ability to recognize equation structures, apply properties of equality systematically, and translate real-world situations into mathematical language. These skills don't just help you solve for $x$; they build the logical reasoning you'll need for systems of equations, inequalities, and eventually quadratics.

Here's the key insight: every technique you learn for solving linear equations comes back to one principle—maintaining balance while isolating the variable. Whether you're clearing fractions, distributing across parentheses, or moving variables from one side to another, you're always applying the same core properties. Don't just memorize steps—understand why each operation keeps the equation true, and you'll be able to tackle any variation the exam throws at you.

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Understanding Equation Structure

Before you can solve an equation, you need to recognize what you're working with. A linear equation contains variables raised only to the first power, which is why its graph is always a straight line.

Definition of a Linear Equation

• First-degree equations contain variables with an exponent of 1—no squares, cubes, or higher powers allowed
• Standard form is written as $Ax + B = C$, where $A$, $B$, and $C$ are constants and $A \neq 0$
• Graphical representation is always a straight line, with each point representing a solution to the equation

Graphing Linear Equations

• Slope-intercept form $y = mx + b$ reveals the slope ($m$) and y-intercept ($b$) immediately
• Two points are all you need to graph any linear equation—plot them and connect with a straight line
• Every point on the line represents a valid solution, making graphs useful for visualizing infinite solution sets

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The Properties That Make It All Work

Every solving technique relies on the properties of equality. These properties guarantee that whatever you do to one side, doing the same to the other keeps the equation true.

Properties of Equality

• Addition/Subtraction properties state that adding or subtracting the same value from both sides preserves equality
• Multiplication/Division properties allow you to multiply or divide both sides by any non-zero number without breaking the balance
• Foundation for all solving—every step you take when isolating a variable uses one of these four properties

Isolating the Variable

• Inverse operations undo what's been done to the variable—add to undo subtraction, multiply to undo division
• Work systematically by moving constants away from the variable term first, then eliminating coefficients
• Balance is everything—perform identical operations on both sides to maintain equality throughout

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Handling Complex Equation Structures

Real exam problems rarely give you simple one-step equations. These techniques help you simplify complicated equations into forms you can solve using basic properties.

Solving Equations with Variables on Both Sides

• Combine like terms on each side first to simplify before moving anything
• Move all variable terms to one side and all constants to the other using addition/subtraction properties
• Isolate the variable once you've consolidated—you're back to a basic one-step or two-step equation

Solving Equations with Parentheses

• Distributive property eliminates parentheses by multiplying each term inside: $a(b + c) = ab + ac$
• Combine like terms after distributing—this often reveals a simpler equation structure
• Then isolate using standard properties of equality once parentheses are cleared

Solving Equations with Fractions and Decimals

• Clear fractions by multiplying every term by the least common denominator (LCD)—this eliminates all denominators at once
• Clear decimals by multiplying by the appropriate power of 10 (×10 for tenths, ×100 for hundredths)
• Standard solving proceeds normally once coefficients become whole numbers

Cross-Multiplication for Proportions

• Proportions in the form $\frac{a}{b} = \frac{c}{d}$ can be solved by cross-multiplying: $a \cdot d = b \cdot c$
• Eliminates fractions instantly—one step converts a proportion into a simple linear equation
• Check denominators first—this method is invalid if any denominator equals zero

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Verification and Application

Solving isn't complete until you've verified your answer and can apply these skills to real situations. These final steps separate correct answers from careless errors.

Checking Solutions

• Substitute back into the original equation—not a simplified version—to verify your answer
• Both sides must equal the same value after substitution; if they don't, retrace your steps
• Catches common errors like sign mistakes, distribution errors, and arithmetic slips

Word Problems Involving Linear Equations

• Define variables clearly—state what $x$ represents in the context of the problem
• Translate phrases into mathematical operations: "more than" means add, "times" means multiply, "is" means equals
• Interpret your answer in context—a negative number of apples doesn't make sense, so check reasonableness

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Quick Reference Table

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Self-Check Questions

1. What do the addition property and multiplication property of equality have in common, and when would you choose one over the other?

2. You encounter the equation $\frac{x + 2}{3} = \frac{5}{6}$. Which method—LCD or cross-multiplication—would be most efficient, and why?

3. Compare solving $3(x - 4) = 15$ versus solving $3x - 4 = 15$. What extra step does the first equation require, and what property justifies it?

4. A student solves an equation and gets $x = -7$, but when checking, the left side equals 12 and the right side equals 10. What does this tell them, and what should they do next?

5. If an FRQ asks you to "set up and solve an equation" for a word problem, what three components must your response include to earn full credit?