An equation is a mathematical statement asserting that two algebraic expressions are equivalent, containing one or more unknown values, typically represented by a variable like $x$. To solve an equation means finding the value(s) of the variable that make the equality true. A unique solution exists when exactly one specific value satisfies the equation, distinguishing it from having no solution or infinitely many solutions.
The Algebraic Process for Linear Equations
Determining if a linear equation has a unique solution involves isolating the variable. A linear equation in one variable, such as $Ax + B = 0$, has a variable with an exponent of one. To begin, remove any parentheses by distributing and combine like terms to simplify the expressions.
The next step uses the properties of equality to gather all variable terms onto one side and all constant terms onto the opposite side. For instance, in $4x + 8 = 18$, subtracting 8 from both sides isolates the variable term, resulting in $4x = 10$.
The final algebraic manipulation involves using the multiplication or division property of equality to make the coefficient of the variable equal to one. Continuing the example, dividing both sides of $4x = 10$ by 4 yields $x = 2.5$. The appearance of the variable isolated on one side and equated to a single, specific real number, like $x = 2.5$, signals that the equation has exactly one unique solution.
Outcomes That Mean No Unique Solution
An equation lacks a unique solution when the algebraic process results in a statement that is always false or always true. This happens when the variable terms completely cancel out on both sides. The resulting statement indicates whether the equation is a contradiction (no solution) or an identity (infinitely many solutions).
If the variables cancel and the remaining statement is a contradiction, such as $0 = 5$, the equation has no solution. This false statement confirms that no value of the variable can make the original equation true. Graphically, this represents two parallel lines that never intersect.
Conversely, if the variables cancel and the remaining statement is a true identity, such as $0 = 0$, the equation has infinitely many solutions. This indicates the original equation is valid for any real number substituted for the variable.
Extending the Concept to Non-Linear Equations
The concept of a unique solution applies to equations where the variable has an exponent greater than one, such as quadratic equations. While a quadratic equation, written as $ax^2 + bx + c = 0$, generally has two solutions, a specific condition must be met for it to have exactly one unique solution.
This condition is determined by the discriminant, the expression found under the square root in the quadratic formula: $b^2 – 4ac$. If the discriminant equals zero, the equation has one unique real solution. For example, in $x^2 + 2x + 1 = 0$, the discriminant is $(2)^2 – 4(1)(1)$, which equals $0$.
When the discriminant is zero, the quadratic formula simplifies to $\frac{-b}{2a}$, yielding only one value for the variable. Graphically, a quadratic equation with one unique solution is represented by a parabola that touches the x-axis at exactly one point.