Solving Linear Equations Revision Notes for Leaving Cert Mathematics

Solving Linear Equations

What is a linear equation?

A linear equation is a mathematical statement that contains an equals sign (=) and shows that two algebraic expressions are equal to each other. The equation contains one or more variables (usually x) raised to the power of 1 only.

Key features:

Contains an equals sign (=)
Variable appears to the power of 1 only
No multiplication of variables together
Forms a straight line when graphed

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Definition: Solving an equation means finding the value of the variable that makes the equation true.

Basic principles for solving linear equations

When solving linear equations, you must follow these fundamental rules to ensure accuracy and maintain mathematical integrity.

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Fundamental Rules for Linear Equations:

Balance principle: Whatever you do to one side of the equation, you must do exactly the same to the other side
Inverse operations: Use opposite operations to "undo" what has been done to the variable
Isolation goal: Your aim is to get the variable by itself on one side of the equation

The balance principle is the foundation of equation solving - it ensures that the equality remains true throughout your solution process.

Step-by-step method for solving linear equations

Simple linear equations

For equations like $2x = 8$ or $x + 3 = 7$ , follow this streamlined approach:

Identify what operation is being performed on the variable
Apply the inverse operation to both sides
Simplify to find the value of the variable
Check your answer by substituting back into the original equation

Multi-step linear equations

For more complex equations, follow this systematic approach:

Remove brackets using the distributive property (if present)
Collect like terms on each side
Move all terms with variables to one side
Move all number terms to the other side
Simplify both sides
Divide or multiply to isolate the variable
Check your solution

Worked examples

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Worked Example 1: Multi-step equation

Solve: $5x - 3 = 2x + 9$

Step 1: Add 3 to both sides $5x - 3 + 3 = 2x + 9 + 3$ $5x = 2x + 12$

Step 2: Subtract 2x from both sides
$5x - 2x = 2x - 2x + 12$ $3x = 12$

Step 3: Divide both sides by 3 $\frac{3x}{3} = \frac{12}{3}$ $x = 4$

Check: Substitute $x = 4$ back into the original equation:

Left side: $5(4) - 3 = 20 - 3 = 17$

Right side: $2(4) + 9 = 8 + 9 = 17$ ✓

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Worked Example 2: Equation with brackets

Solve: $5(2x - 4) = 3(2x - 1) - 1$

Step 1: Remove brackets using the distributive property $10x - 20 = 6x - 3 - 1$

Step 2: Simplify the right side $10x - 20 = 6x - 4$

Step 3: Add 20 to both sides $10x - 20 + 20 = 6x - 4 + 20$ $10x = 6x + 16$

Step 4: Subtract 6x from both sides $10x - 6x = 6x - 6x + 16$ $4x = 16$

Step 5: Divide both sides by 4 $x = 4$

Check: Substitute $x = 4$ back into the original equation to verify.

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Worked Example 3: Simple one-step equation

Solve: $3x = 15$

Step 1: Divide both sides by 3

$\frac{3x}{3} = \frac{15}{3}$

$x = 5$

Types of linear equations you'll encounter

Understanding the different types of linear equations helps you choose the most efficient solution method:

One-step equations

$2x = 8$ (multiply/divide)
$x + 5 = 12$ (add/subtract)

Two-step equations

$2x - 3 = 7$ (add then divide)
$\frac{x}{4} + 1 = 6$ (subtract then multiply)

Multi-step equations

$5x + 2 = 3x - 4$ (variables on both sides)
$2(x + 3) = x + 10$ (brackets to expand)

Applications in geometry

Linear equations frequently appear in geometric contexts, providing practical applications for algebraic problem-solving skills.

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Common Geometric Applications:

Linear equations are particularly useful when working with:

Perimeter problems where sides are expressed algebraically
Angle calculations using known geometric properties
Area relationships involving unknown dimensions

Perimeter problems

When shapes have sides expressed algebraically, you can form equations to solve for unknown values.

For the triangle and rectangle shown:

Triangle perimeter: $(x + 1) + (x + 2) + 2x = 4x + 3$
Rectangle perimeter: $2x + 2(30 - 2x) = 2x + 60 - 4x = 60 - 2x$

Angle problems

In triangles, the sum of angles equals 180°, creating linear equations.

For the triangle with angles $(3x - 4)°$ , $(2x - 6)°$ , and $(50 - x)°$ : $(3x - 4) + (2x - 6) + (50 - x) = 180$

Simplifying: $4x + 40 = 180$ Therefore: $4x = 140$ , so $x = 35$

Common exam tips and traps

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Exam Success Tips:

Always show your working - marks are awarded for method
Check your answer by substituting back into the original equation
Be careful with signs when moving terms across the equals sign
Use brackets when substituting negative values during checking

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Common Traps to Avoid:

Sign errors: When moving $-3x$ from left to right, it becomes $+3x$
Forgetting to distribute: $2(x + 3) = 2x + 6$ , not $2x + 3$
Dividing incorrectly: When you have $-2x = 8$ , remember $x = -4$
Not checking: Always verify your solution makes the original equation true

Key formulas and rules

Understanding these fundamental mathematical rules will support your equation-solving process:

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Essential Formulas and Rules:

Distributive property: $a(b + c) = ab + ac$

Balance rule: If $a = b$ , then $a + c = b + c$ and $a - c = b - c$

Multiplication rule: If $a = b$ , then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$ (where $c ≠ 0$ )

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Key Points to Remember:

Linear equations contain variables to the power of 1 only and form straight lines when graphed
Always maintain balance - what you do to one side, do to the other side
Work systematically - remove brackets, collect like terms, move variables to one side, solve
Check your answer by substituting back into the original equation
In geometry applications, use known properties like triangle angles summing to 180° or perimeter formulas to create your equations

Solving Linear Equations (Leaving Cert Mathematics): Revision Notes