# Linear Equations in One Variable

**Linear equation in one variable** is the equation with standard form $$${\color{purple}{{{m}{x}+{b}={0}}}}$$$.

$$$m$$$ and $$$b$$$ are some numbers and $$$x$$$ is a variable.

Examples of linear equations are:

- $$${4}{x}+{2}={0}$$$
- $$$-{2}{a}-{3}={0}$$$
- $$$\frac{{3}}{{2}}{m}-\frac{{5}}{{3}}={0}$$$

Using equivalence of equations, we can convert some other equations into the standard form:

- $$${2}{x}={5}$$$ is equivalent to $$${2}{x}-{5}={0}$$$ (subtract 5 from both sides of equation)
- $$$\frac{{3}}{{2}}{x}={5}-\frac{{2}}{{3}}{x}$$$ becomes $$$\frac{{13}}{{6}}{x}-{5}={0}$$$ (move everything to the left and combine like terms)
- $$$\sqrt{{{2}}}{x}-{5}={x}+{2}$$$ becomes $$${\left(\sqrt{{{2}}}-{1}\right)}{x}-{7}={0}$$$ (move everything to the left and combine like terms)
- $$$\frac{{1}}{{y}}={2}$$$ becomes $$$-{2}{y}+{1}={0}$$$ (multiply both sides by $$${y}$$$ and move everything to the left)

**Equation is linear**, when it is written in standard form and variable is raised to the first power only.

Following are **NOT** linear equations:

- $$${2}{{x}}^{{2}}+{3}={0}$$$ (variable raised to the second power)
- $$${2}{y}-{3}=\frac{{3}}{{2}}{{y}}^{{2}}$$$ (there is variable, raised to the second power)
- $$$\frac{{1}}{{y}}+{y}={2}$$$ (if we multiply both sides by $$${y}$$$, then we will get $$${1}+{{y}}^{{2}}={2}{y}$$$, which is not quadratic)

**Exercise 1.** Determine, whether $$${2}{x}=-{5}$$$ is linear and write it in standard form if it is.

**Answer**: yes; $$${2}{x}+{5}={0}$$$.

**Exercise 2.** Determine, whether $$${1}=\frac{{2}}{{3}}{a}$$$ is linear and write it in standard form if it is.

**Answer**: yes; $$$\frac{{2}}{{3}}{a}-{1}={0}$$$.

**Exercise 3.** Determine, whether $$${{x}}^{{2}}={7}$$$ is linear and write it in standard form if it is.

**Answer**: no.

**Exercise 4.** Determine, whether $$$\frac{{1}}{{x}}+{5}={x}$$$ is linear and write it in standard form if it is.

**Answer**: no. Multiplying both sides by $$${x}$$$ gives $$${1}+{5}{x}={{x}}^{{2}}$$$.

**Exercise 5.** Determine, whether $$$\frac{{3}}{{x}}=\frac{{7}}{{3}}$$$ is linear and write it in standard form.

**Answer**: yes; $$$\frac{{7}}{{3}}{x}-{3}={0}$$$.