Linear equations with one, none or infinitely many solutions Knowledge-Based Study

Study notes

1. The Nature of Linear Equations

   A linear equation in one variable can be written in the form $ax + b = cx + d$, where $a$, $b$, $c$, and $d$ are real numbers. When solving such an equation, our goal is to isolate the variable $x$ by applying inverse operations systematically.

   Unlike quadratic or higher-degree equations which may have multiple distinct solutions, a standard single-variable linear equation falls into exactly one of three distinct categories:

   • One unique solution: Exactly one value of the variable makes the equation true.
   • No solution: No value of the variable can make the equation true.
   • Infinitely many solutions: Every real number makes the equation true.

   Understanding the algebraic structure of an equation allows us to determine which outcome applies even before completing every step of isolation.

   Conditional Equations

   An equation has exactly one solution when the coefficients of the variable on each side of the equation are different after both sides are fully simplified.

   Consider an equation simplified to the form:

   $$ax = b$$

   If $a \neq 0$, dividing both sides by $a$ gives the single, unique solution $x = \frac{b}{a}$. Such an equation is called a conditional equation because it is true on the condition that $x$ equals that specific value.

   Worked Example

   Solve the equation:

   $$3(2x - 4) = 2x + 8$$

   1. Distribute: Expand the left-hand side.
      $6x - 12 = 2x + 8$
   2. Collect variable terms: Subtract $2x$ from both sides.
      $4x - 12 = 8$
   3. Collect constant terms: Add 12 to both sides.
      $4x = 20$
   4. Isolate the variable: Divide both sides by 4.
      $x = 5$

   Since the variable coefficients were not equal ($6 \neq 2$), the variable did not eliminate, resulting in exactly one solution: $x = 5$.

   Inconsistent Equations

   An equation has no solution when simplifying leads to the variable terms eliminating completely, leaving a statement that is mathematically false (a contradiction), such as $0 = 5$ or $-3 = 4$.

   In standard form, if an equation reduces to:

   $$0x = k \quad \text{where } k \neq 0$$

   there is no real number that can be multiplied by 0 to yield a non-zero number $k$. Therefore, no value of $x$ can satisfy the statement. The solution set is empty, often denoted by the symbol $\emptyset$.

   Worked Example

   Solve the equation:

   $$4(x + 2) - 2x = 2(x + 5)$$

   1. Expand and simplify both sides:
      Left side: $4x + 8 - 2x = 2x + 8$
      Right side: $2x + 10$
   2. Compare the sides:
      $2x + 8 = 2x + 10$
   3. Subtract $2x$ from both sides:
      $8 = 10$

   The statement $8 = 10$ is impossible and false for any value of $x$. Consequently, the equation has no solution.

   Identities

   An equation has infinitely many solutions when simplifying leads to an expression that is identically equal on both sides, or when the variable terms eliminate to produce a universally true statement, such as $0 = 0$ or $7 = 7$.

   Such an equation is known as an identity because it holds true for every real value substituted for the variable.

   In reduced form:

   $$0x = 0$$

   Because any real number multiplied by 0 results in 0, every real number is a valid solution. The solution set is all real numbers, denoted as $\mathbb{R}$.

   Worked Example

   Solve the equation:

   $$5(x - 1) + 3 = 2(2x - 1) + x$$

   1. Expand both sides:
      Left side: $5x - 5 + 3 = 5x - 2$
      Right side: $4x - 2 + x = 5x - 2$
   2. Compare both sides:
      $5x - 2 = 5x - 2$
   3. Subtract $5x$ from both sides:
      $-2 = -2$ (or add 2 to obtain $0 = 0$)

   Because both sides are identical expressions, the equation is satisfied by any real number. Hence, there are infinitely many solutions.

   How to Classify Equations Quickly

   To determine the number of solutions of any linear equation $Ax + B = Cx + D$ without completing all final algebraic steps, compare the variable coefficients ($A$ and $C$) and the constant terms ($B$ and $D$):

   • One Solution: The coefficients of $x$ are different ($A \neq C$). The constants $B$ and $D$ can be equal or different. The lines intersect at one point.
   • No Solution: The coefficients of $x$ are equal ($A = C$), but the constant terms are different ($B \neq D$). Subtracting the variable term leaves a false statement like $B = D$. Graphically, these represent distinct parallel lines that never intersect.
   • Infinitely Many Solutions: Both the coefficients of $x$ and the constant terms are equal ($A = C$ and $B = D$). Graphically, both sides represent the exact same line.