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# Solving a linear equation

We solve a linear equation by applying the same operations to both sides of the expression. The objective is to isolate the unknown $x$.

Let's start with an example. We want to solve $$2\Tred{x}-1 = 3.$$

• We add $\Tgreen{1}$ to both sides $$2\Tred{x} = 2\Tred{x}-1 + \Tgreen{1} = 3+\Tgreen{1} = 4$$
• We divide both sides by $\Tgreen{2}$ $$\Tred{x} = \frac{2\Tred{x}}{\Tgreen{2}} = \frac{4}{\Tgreen{2}} = 2$$
• This means that the solution is $\Tred{x} = 2$.

We can replicate the reasoning to solve a general linear equation $$\Tblue{a} \Tred{x} + \Tblue{b} = \Tblue{c}$$ for numbers $\Tblue{a}\ne0$, $\Tblue{b}$ and $\Tblue{c}$. This gives the solution $\displaystyle \Tred{x} = \frac{\Tblue{c-b}}{\Tblue{a}}.$

• We subtract $\Tgreen{b}$ from both sides: $\Tblue{a}\Tred{x} = \Tblue{a}\Tred{x} + \Tblue{b} - \Tgreen{b} = \Tblue{c} - \Tgreen{b}$
• We divide both sides by $\Tgreen{a}$: $\displaystyle \Tred{x} = \frac{\Tblue{a}\Tred{x}}{\Tgreen{a}} = \frac{\Tblue{c-b}}{\Tgreen{a}}.$