More complex linear equations may involve fractions, brackets or several fractional terms on one or both sides. The aim is still to find the value of the variable that makes the equation true, but extra care is needed to keep the working clear and accurate.

A systematic approach is essential.

 

 

Understanding the Structure

Complex linear equations may include:
• fractions in one or more terms
• the variable appearing several times
• brackets that need expanding

 

A common and efficient strategy is to clear fractions early by multiplying through by a common denominator.

 

 

Clearing Fractions from an Equation

Consider the equation:

$$
\frac{x}{3} + \frac{x}{2} = 5
$$

 

The common denominator of \( 3 \) and \( 2 \) is \( 6 \). Multiply every term by \( 6 \):

$$
6\left(\frac{x}{3}\right) + 6\left(\frac{x}{2}\right) = 6 \times 5
$$

 

Simplify:

$$
2x + 3x = 30
$$

$$
5x = 30
$$

 

Divide by \( 5 \):

$$
x = 6
$$

 

 

Solving Equations with Several Fractional Terms

Consider:

$$
\frac{2x}{5} + \frac{3}{4} = \frac{x}{2}
$$

 

The common denominator of \( 5 \), \( 4 \) and \( 2 \) is \( 20 \). Multiply through by \( 20 \):

$$
20\left(\frac{2x}{5}\right) + 20\left(\frac{3}{4}\right) = 20\left(\frac{x}{2}\right)
$$

 

Simplify:

$$
8x + 15 = 10x
$$

 

Subtract \( 8x \) from both sides:

$$
15 = 2x
$$

 

Divide by \( 2 \):

$$
x = \frac{15}{2}
$$

 

 

Equations with Fractions and Brackets

Consider:

$$
\frac{1}{3}(2x + 4) = \frac{x}{2} - 1
$$

 

The common denominator is \( 6 \). Multiply through by \( 6 \):

$$
2(2x + 4) = 3x - 6
$$

 

Expand the bracket:

$$
4x + 8 = 3x - 6
$$

 

Subtract \( 3x \) from both sides:

$$
x + 8 = -6
$$

 

Subtract \( 8 \):

$$
x = -14
$$

 

 

Forming and Solving an Equation from a Worded Problem

For example:

“One third of a number plus one half of the same number is equal to 10.”

 

This can be written as:

$$
\frac{x}{3} + \frac{x}{2} = 10
$$

 

Clearing fractions using a common denominator of \( 6 \):

$$
2x + 3x = 60
$$

$$
5x = 60
$$

$$
x = 12
$$

 

 

Key Points to Remember

Clear fractions by multiplying every term by a common denominator.
Expand brackets before collecting like terms.
Keep working organised and step by step.
Check the solution by substitution if possible.

 

Solving complex linear equations relies on careful manipulation, but the underlying principles are the same as for simpler equations.