## Solving Linear Equations

The simplest equation to solve is a linear equation. A linear equation is an equation where the highest exponent of the variable is \(\text{1}\). The following are examples of linear equations:

\begin{align*} 2x + 2 & = 1 \\ \cfrac{2 – x}{3x + 1} & = 2 \\ 4(2x – 9) – 4x & = 4 – 6x \\ \cfrac{2a – 3}{3} – 3a & = \cfrac{a}{3} \end{align*}

Solving an equation means finding the value of the variable that makes the equation true. For example, to solve the simple equation \(x + 1 = 1\), we need to determine the value of \(x\) that will make the left hand side equal to the right hand side. The solution is \(x = 0\).

The solution, also called the root of an equation, is the value of the variable that satisfies the equation. For linear equations, there is at most one solution for the equation.

To solve equations we use algebraic methods that include expanding expressions, grouping terms, and factorising.

For example:

\begin{align*} 2x + 2 & = 1 \\ 2x & =1 – 2 \quad \text{ (rearrange)} \\ 2x & = -1 \quad \text{ (simplify)} \\ x & = -\cfrac{1}{2} \quad \text{(divide both sides by } 2\text{)} \end{align*}

Check the answer by substituting \(x=-\cfrac{1}{2}\).

\begin{align*} \text{LHS } & = 2x + 2 \\ & = 2(-\cfrac{1}{2}) + 2 \\ & = -1 + 2 \\ & = 1 \\ \text{RHS } & =1 \end{align*}

Therefore \(x=-\cfrac{1}{2}\)

The following video gives an introduction to solving linear equations.