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:

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; $\frac{{7}}{{3}}{x}-{3}={0}$ .