# Linear equation with fractions

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

1. If present, remove parentheses and denominators.
2. Group the variable terms in one member and the independent terms in the other.
3. Reduce like terms.
4. 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