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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 $$\Tgreen{a}$$: $$\displaystyle \Tred{x} = \frac{\Tblue{a}\Tred{x}}{\Tgreen{a}} = \frac{\Tblue{c-b}}{\Tgreen{a}}.$$