Solving Linear Equations With Fractions Revision Notes for Leaving Cert Mathematics

Solving Linear Equations With Fractions

Linear equations containing fractions require a systematic approach to eliminate the fractions and solve for the unknown variable. The key principle is to clear the fractions first by multiplying through by appropriate values, then solve the resulting equation using standard algebraic techniques.

infoNote

Systematic Approach to Fraction Equations

The fundamental strategy for solving linear equations with fractions is to eliminate all fractions early in the process. This transforms the problem into a standard linear equation that can be solved using familiar algebraic techniques.

Method for single fraction equations

When dealing with a linear equation containing one fraction, we multiply both sides of the equation by the denominator to eliminate the fraction.

lightbulbExample

Worked Example: Single Fraction Equation

Solve $\frac{2x - 1}{5} = 3$

Step-by-step solution:

Multiply both sides by 5 to clear the fraction: $\frac{5(2x - 1)}{5} = 3 \times 5$
Simplify: $2x - 1 = 15$
Add 1 to both sides: $2x - 1 + 1 = 15 + 1$ $2x = 16$
Divide both sides by 2: $x = 8$

Method for multiple fraction equations

When an equation contains more than one fraction, we multiply each term by the lowest common multiple (LCM) of all denominators. This eliminates all fractions in one step.

infoNote

Why Use the LCM?

Using the LCM ensures that all fractions are completely eliminated in a single multiplication step, avoiding partial fractions and making the subsequent algebra much simpler.

lightbulbExample

Worked Example 1: Multiple Fractions

Solve: $\frac{4x}{5} - \frac{x}{2} = \frac{3}{4}$

Solution:

Find the LCM of denominators 5, 2, and 4: LCM(5, 2, 4) = 20
Multiply each term by 20: $\frac{20(4x)}{5} - \frac{20(x)}{2} = \frac{20(3)}{4}$
Simplify each fraction: $4(4x) - 10(x) = 5(3)$ $16x - 10x = 15$
Combine like terms: $6x = 15$
Solve for x: $x = \frac{15}{6} = \frac{5}{2} = 2\frac{1}{2}$

lightbulbExample

Worked Example 2: Brackets in Fractions

Solve: $\frac{x + 4}{3} - \frac{x + 2}{4} = \frac{7}{6}$

Solution:

Find the LCM of denominators 3, 4, and 6: LCM(3, 4, 6) = 12
Multiply each term by 12: $\frac{12(x + 4)}{3} - \frac{12(x + 2)}{4} = \frac{12(7)}{6}$
Simplify each fraction: $4(x + 4) - 3(x + 2) = 2(7)$
Expand brackets: $4x + 16 - 3x - 6 = 14$
Combine like terms: $x + 10 = 14$
Subtract 10 from both sides: $x = 4$

Key rules and exam tips

infoNote

Essential Rules for Success

Always identify all denominators before starting to solve
Find the LCM carefully - this is crucial for clearing all fractions effectively
Multiply every term in the equation by the LCM, not just the fractions
Check your answer by substituting back into the original equation
Show all working clearly - examiners award marks for method even if the final answer is incorrect

Common mistakes to avoid

chatImportant

Common Pitfalls to Avoid

Forgetting to multiply constant terms by the LCM
Incorrectly calculating the LCM of the denominators
Making errors when expanding brackets after clearing fractions
Not maintaining equation balance when performing operations

These mistakes are frequently seen in exams and can be easily avoided with careful, systematic working.

bookmarkSummary

Key Points to Remember

Clear fractions first by multiplying through by the appropriate value
For single fractions: multiply both sides by the denominator
For multiple fractions: find the LCM of all denominators and multiply every term
Always check your answer by substituting back into the original equation
Show all steps clearly in your working for maximum marks in exams

Solving Linear Equations With Fractions (Leaving Cert Mathematics): Revision Notes