Solving linear equations involving fractions Revision Notes for AQA GCSE Further Maths

Solving linear equations involving fractions

What you need to know first

Before tackling linear equations with fractions, you should be comfortable with the basic mathematical operations of addition, subtraction, multiplication, and division. These fundamental skills form the foundation for working with more complex fractional equations.

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If you're not confident with basic fraction operations like adding, subtracting, multiplying, and dividing fractions, it's worth reviewing these skills first before attempting linear equations with fractions.

Understanding the approach

When solving linear equations that contain fractions, the most effective strategy is to eliminate the fractions entirely. This makes the equation much easier to work with and reduces the chance of making errors.

The key technique involves finding the Least Common Multiple (LCM) of all denominators in the equation, then multiplying every term in the equation by this LCM. This process clears all the fractions and leaves you with a simpler linear equation to solve.

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Think of this approach as "clearing the clutter" - by removing fractions, you transform a complex-looking equation into something much more manageable to work with.

Step-by-step method

Let's work through a complete example to see how this method works in practice.

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Worked Example: Solving a Linear Equation with Fractions

Solve the equation $\frac{x + 2}{6} = \frac{x - 6}{2}$

Step 1: Identify the denominators In this equation, we have denominators of 6 and 2.

Step 2: Find the LCM The LCM of 6 and 2 is 6, since 6 is the smallest number that both 6 and 2 divide into evenly.

Step 3: Multiply every term by the LCM Multiply both sides of the equation by 6:

$6 \times \frac{x + 2}{6} = 6 \times \frac{x - 6}{2}$

This gives us: $(x + 2) = 3(x - 6)$

Step 4: Expand and simplify Expanding the right side: $x + 2 = 3x - 18$

Step 5: Collect like terms Rearranging to get all x terms on one side: $x + 2 = 3x - 18$ $2 + 18 = 3x - x$ $20 = 2x$

Step 6: Solve for x Dividing both sides by 2: $x = 10$

Important principle to remember

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When you multiply a fraction by a whole number, you only multiply the numerator (the top part of the fraction). The denominator stays the same.

This is why when we multiplied $\frac{x - 6}{2}$ by 6, we got $6 \times \frac{x - 6}{2} = 3(x - 6)$ , not $\frac{6(x - 6)}{12}$ .

Why this method works

The reason we multiply by the LCM is that it's the most efficient way to clear all fractions at once. By choosing the LCM, we ensure that when we multiply each fraction, the denominators cancel out completely, leaving us with whole number coefficients that are easier to work with.

This approach is much more reliable than trying to work with fractions throughout the entire solution process, where small errors can easily compound and lead to incorrect answers.

Common exam tips

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Essential Tips for Success:

Always start by identifying all the denominators in your equation
Find the LCM carefully - this is a crucial step that determines the success of your solution
When multiplying through by the LCM, make sure you multiply every single term, not just the fractions
Check your answer by substituting it back into the original equation
Be extra careful with negative signs when expanding brackets

Remember!

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Key Points to Remember:

The most effective way to solve linear equations with fractions is to eliminate the fractions by multiplying through by the LCM of all denominators
When multiplying a fraction by a whole number, only the numerator gets multiplied
Always multiply every term in the equation by the LCM, not just the fractional terms
The LCM method transforms a complex fractional equation into a simpler linear equation
Always check your final answer by substituting back into the original equation

Solving linear equations involving fractions (AQA GCSE Further Maths): Revision Notes