Linear Equations: Practice Questions & Notes

A linear equation is an algebraic equation in which the highest power of the variable(s) is 1.
The general form of a linear equation in one variable is:

ax + b = 0

Where:

$a \neq 0$
$x$ is the variable
$a, b$ are constants

Linear equations represent straight lines when plotted on a graph (in two variables).

Intuitive Understanding

Think of a linear equation as a balance scale:

You can add/subtract/multiply/divide both sides as long as you maintain the balance.
Your goal is to isolate the variable on one side.

Solving a linear equation is basically the process of finding the value of the unknown that makes both sides equal.

Types of Linear Equations

One Variable

Form: $ax + b = 0$
Example: $3x + 5 = 11$

Two Variables

Form: $ax + by + c = 0$
Example: $2x + 3y = 7$

Three Variables

Form: $ax + by + cz = d$
Used in systems of equations.

Key Concepts & Formulas

General Form (1 variable)

ax + b = 0 \Rightarrow x = -\frac{b}{a}

Operations That Preserve Equality

You can:

Add/subtract same value from both sides
Multiply/divide both sides by same non-zero number

Solving Linear Equations: Steps

For One Variable

\text{Step 1:} \ \text{Expand brackets (if any)} \\ \text{Step 2:} \ \text{Move all variables to one side and constants to the other} \\ \text{Step 3:} \ \text{Simplify and isolate the variable}

For Two Variables

You’ll usually be solving a system of two equations. Use:

Substitution method
Elimination method
Cross multiplication (for quick solving)
Graphical method (conceptual)

Conceptual Tips

Pitfall	Tip
Forgetting to flip sign when moving a term	Always change sign when moving terms across the equals sign
Dividing by zero	Never allowed. Always check $a \neq 0$
Not simplifying both sides	Always reduce terms before solving
Ignoring negative coefficients	Be mindful during operations; sign errors are common
Misinterpreting word problems	Translate English to math carefully—identify variables clearly

Linear Equations in Word Problems

Common Phrases → Math Translation

Phrase	Equation Form
"Sum of a number and 5 is 12"	$x + 5 = 12$
"Twice a number is 10"	$2x = 10$
"A number decreased by 4 is 9"	$x - 4 = 9$
"Three times a number plus 1 equals 10"	$3x + 1 = 10$

Linear Equations in Two Variables

Form: $ax + by = c$
Solution is not a fixed number, but an infinite set of pairs $(x, y)$ that satisfy the equation.

To get a unique solution, you need two such equations.

Methods to Solve 2 Variable Equations

1. Substitution Method

Solve one equation for one variable
Substitute into the other

2. Elimination Method

Multiply equations to align coefficients
Add/subtract equations to eliminate one variable

3. Cross Multiplication Method (when equations are in standard form)

For equations:

a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2

Use:

\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}

Visual Understanding

1 Variable:

Represents a point on a number line.

2 Variables:

Each equation is a line on a graph.

One solution: Lines intersect
Infinite solutions: Lines overlap
No solution: Lines are parallel

Examples

Example 1: One Variable

Q: Solve $2x - 5 = 9$
A:

2x = 14 \Rightarrow x = 7

Example 2: Word Problem

Q: The sum of a number and its double is 18. Find the number.
A:
Let the number be $x$ . Then:

x + 2x = 18 \Rightarrow 3x = 18 \Rightarrow x = 6

Example 3: Two Variables

Solve:

x + y = 10 \\ x - y = 4

Add:

2x = 14 \Rightarrow x = 7 \\ \Rightarrow y = 3

Example 4: Fractions

Q: Solve $\frac{3x - 4}{2} = 5$
A:
Multiply both sides by 2:

3x - 4 = 10 \Rightarrow x = \frac{14}{3}