Category: Lineare Gleichungen

Definition of Variable

The properties of numbers are often written with the help of letters.

For example,we can write the commutative property of addition as ${a}+{b}={b}+{a}$.

In this equality we can plug any numbers instead of $a$ and $b$: ${32}+{4573}={4573}+{32}$, ${350}+{3}={3}+{350}$ etc.

Definition of Equation. Roots of the Equation.

The equation is two expressions separated by an equal sign (=). We will mainly deal with equations that contain one or more variables.

Examples of equations:

• ${2}+{3}={5}$ (no variables)
• ${4}{x}+{3}={1}$ (variable in left expression)
• ${15}={{x}}^{{2}}+{2}{x}$ (variable in right expression)
• ${{x}}^{{3}}+{4}-{2}{{x}}^{{2}}=\frac{{1}}{{x}}+{{2}}^{{x}}$ (variable in both expressions)

Roots of the equation are such values of the variable, that turn equation into correct equality.

Equivalent Equations

Equations, that have same roots are called equivalent. In particular, equations that don't have roots are equivalent.

Example 1. Equations ${x}+{2}={7}$ and ${2}{x}+{1}={11}$ are equivalent, because each of them has unique root 5.

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:

One-Step Linear Equations

One-step linear equation is an equation, that requires only one action (operation) to be solved.

Basically, there are 4 kinds of such equations:

• ${x}+{3}={7}$ (solve by using subtraction)
• ${a}-{2}=-{3}$ (solve by using addition)
• ${2}{y}={1}$ (solve by using division)
• $\frac{{1}}{{4}}{x}={5}$ (solve by using multiplication)

You don't need to remember above kinds of equation. Just keep in mind, that to solve linear equation, we need to isolate variable, i.e. keep it on one side of equation and move everything else to another side.

Two-Step Linear Equations

Two-step linear equation is an equation, that requires two actions (operations) in order to be solved.

Basically, there are 2 kinds of such equations:

• ${2}{x}+{5}={15}$ (add/subtract, then multiply/divide)
• $\frac{{{a}-{2}}}{{4}}={3}$ (multiply/divide, then add/subtract)

Above 2 kinds of linear equations are general. Addition/subtraction, multiplication/division can come in any order. Luckily, you don't need to remember all possible situations. Just keep in mind, that to solve linear equation, we need to isolate variable, i.e. keep it on one side of equation and move everything else to another side.

Multi-Step Linear Equations

Multi-step linear equation is an equation, that requires more than two actions (operations) in order to be solved.

Such equations are solved in a same manner as one-step linear equations and two-step linear equations, using properties of expressions, except you, probably, need to do many operations.

Absolute Value Equations

Absolute value equations are equations that contain an absolute value sign.

Example 1. Solve the following equation: ${\left|{x}\right|}=5$.

We know, that absolute value makes any number positive.