Equations

What is an equation?

An equation is a mathematical statement asserting that two algebraic expressions are equal. These expressions contain one or more unknowns (variables), and solving the equation means finding all the values of the unknowns that make the equality true.

For example, the following equation is satisfied when x = 1:

$$ 3x - 1 = 2x $$

Verification. Substituting x=1 into the equation gives: $$ 3 \cdot 1 = 2 \cdot 1 $$ which confirms that the equality holds.

The expression on the left of the equals sign is called the left-hand side (LHS), and the one on the right is the right-hand side (RHS).

The values of the variable that satisfy the equality are called the solutions or roots of the equation. In this case, x=1 is a solution.

To solve an equation means to determine the entire set of values-the solution set-that satisfies the equality.

In the example above, the solution set consists of a single element: $$ S = \{1\} $$

Types of Equations by Number of Solutions

• Determinate (proper) equations
  These have a finite number of solutions.

  Example. The equation $$ 2x = 4 $$ has a unique solution: $$ x = \frac{4}{2} = 2 $$ Since there is a finite solution set (just one solution), the equation is determinate.

• Indeterminate equations
  These are true for infinitely many values of the variable.

  Example. The equation $$ 0 \cdot x = 0 $$ holds for every value of $x$, so it has infinitely many solutions.

• Impossible equations
  These admit no solution.

  Example. Consider $$ x + 1 = x + 3 $$ Subtracting $x$ from both sides gives: $$ 1 = 3 $$ which is a contradiction. Hence, the equation has no solution and is called impossible.

Standard Form of an Equation

An equation is in standard form (or canonical form) when it is written as

$$ P(x) = 0 $$

where P is a simplified polynomial without like terms.

For example, the equation 3x-1=2x can be rewritten as:

$$ 3x - 1 = 2x $$

$$ 3x - 1 - 2x = 2x - 2x $$

$$ x - 1 = 0 $$

Here, the constant term (the term without the variable) is -1.

Example. In this case, the constant term is -1: $$ x \color{red}{-1} = 0 $$

The degree of an equation is the degree of its polynomial in standard form, i.e. the highest exponent of the variable.

Example. This equation is first degree because the highest exponent of $x$ is 1: $$ x^{\color{red}1} - 1 = 0 $$ First-degree equations are also known as linear equations.

The Domain of an Equation

When solving an equation, it is essential to know the set of numbers in which we are looking for solutions. This set is called the domain (or set of definition) of the equation.

By default, unless otherwise specified, the domain is assumed to be the set of real numbers R. But the domain can change depending on the context, and it can completely alter the outcome of the problem.

Example

Consider the equation: $$ 2x = 1 $$

- Over the real numbers R, the solution is: $$ x = \tfrac{1}{2} $$

- Over the natural numbers N, there is no solution, since no natural number multiplied by 2 equals 1.

This shows why specifying the domain is crucial: it prevents ambiguity and ensures clarity in problem-solving.

Example 2

Consider the equation: $$ \frac{1}{x-3} = 2 $$

Here, $x=3$ must be excluded from the domain because it would make the denominator zero. Thus, the domain is: $$ R \setminus \{3\} $$

Types of Equations

Equations can be classified in several ways. For instance:

• Polynomial equations
  Equations in which the variable appears only in the numerator.

  Example. $$ \frac{x+1}{2} = 0 $$

• Fractional equations
  Equations in which the variable also appears in the denominator.

  Example. $$ \frac{x+1}{2x} = 0 $$

• Literal (parametric) equations
  Equations that involve not only unknowns and constants but also parameters-letters representing fixed, unspecified values.

  Example. $$ \frac{kx+1}{2} = 0 $$ where k is a parameter and x is the unknown.

Other important families of equations include:

• Polynomial equations
• Linear equations (first degree)
• Quadratic equations (second degree)
• Absolute value equations
• Exponential equations
• Irrational equations
• Trigonometric equations
• Logarithmic equations

Remarks

• Equations vs. Identities
  An equation is an identity if the equality holds for every possible value of the variable. Otherwise, it is just an equation, valid only for specific values (or for none).

  Example. The equation $$ 2x = 6 $$ is true only when $x=3$: $$ x = \frac{6}{2} = 3 $$ Since it does not hold for all values, it is an equation, not an identity. On the other hand, $$ (x+1)^2 = x^2 + 2x + 1 $$ is always true, no matter the value of x. For instance: $$ x=0 \ \Rightarrow \ (0+1)^2 = 1 $$ $$ x=2 \ \Rightarrow \ (2+1)^2 = 9 $$ $$ x=-3 \ \Rightarrow \ (-3+1)^2 = 4 $$ Because the equality holds universally, this is an identity.