A linear equation is a mathematical equality between two algebraic expressions, called members, in which known and unknown elements (called variables) appear, and that only involves adding and subtracting a variable to the first power. For example, **x + 3 = 4 - 5x **is a **Linear equation** or of **first grade**.

**Linear equation with fractions**

It can be written in the form ax / b = c / d, where a, b, c and d are real numbers, with a ≠ 0, b ≠ 0 and d ≠ 0. For example: 15x / 2 = 7/8.

**Solving linear equations with fractions**

- If present, remove parentheses and denominators.
- Group the variable terms in one member and the independent terms in the other.
- Reduce like terms.
- Clear the variable.

**Example 1**: Solve the following equation x / 2 + 1/3 = 5.

First, we add 1/3 on both sides of the equality:

x / 2 + 1/3 - 1/3 = 5 - 1/3

x / 2 = 5 - 1/3

x / 2 = (8 - 1) / 3

x / 2 = 7/3

Since we have grouped the variable in one member and the independent terms in the other, we clear x. To do this we multiply both sides of equality by 2:

2 · (x / 2) = (7/3) · 2

Considering that the multiplication of fractions is linear, numerator by numerator and denominator by denominator, we have:

x = 14/3

And this is the solution value to the linear equation. To check it, we substitute it in it and check if equality is met:

x / 2 + 1/3 = 5

(14/3) + 1/3 = 5

(14 + 1)/3 = 5

5 = 5

**Example 2**: Solve the following equation [(4x - 3) / 2] - (5x / 6) = (3/10) + x

We solve both sides of equality as any fraction:

[6 (4x - 3) - 10x] / 12 = (3 + 10x) / 10

(24x - 18 - 10x) / 12 = (3 + 10x) / 10

(14x - 18) / 12 = (3 + 10x) / 10

10 (14x - 18) = 12 (3 + 10x)

140x - 180 = 36 + 120x

140x - 120x = 36 + 180

20x = 216

x = 216/20

We simplify:

x = 54/5