One solution

Definition

One solution refers to a scenario in which a linear equation in one variable has exactly one unique answer. This means that when the equation is solved, it produces a specific value for the variable, indicating that there is a single point of intersection between the line represented by the equation and the x-axis. Understanding this concept is essential for analyzing equations and their applications, as it helps in determining the relationship between variables and identifying feasible solutions in real-world situations.

5 Must Know Facts For Your Next Test

1. For a linear equation in one variable to have one solution, the coefficients of the variable must not create a contradiction, like 0 = 5.
2. Graphically, one solution indicates that the line crosses the x-axis at exactly one point, meaning there is only one x-value that satisfies the equation.
3. If an equation simplifies to a true statement (like $$x = 3$$), it confirms that there is one specific solution.
4. One solution occurs in equations where the variable term has a non-zero coefficient, ensuring that it can be isolated to find a unique answer.
5. In real-world applications, one solution often represents a specific scenario, such as finding a single price point that satisfies supply and demand.

Review Questions

• What does it mean for a linear equation to have one solution, and how can this be identified?
  • A linear equation has one solution when it can be simplified to yield a unique value for the variable. This means that when you isolate the variable through algebraic manipulation, you arrive at an equation like $$x = 4$$. Identifying this involves ensuring that the coefficients do not lead to contradictions or undefined expressions.
• How does having one solution in an equation relate to graphing its linear representation?
  • Having one solution corresponds to the graphical representation of a linear equation intersecting the x-axis at exactly one point. This indicates that there is only one x-value for which the equation holds true. The slope and intercept of the line will determine where it crosses the x-axis, reinforcing that unique relationship between x and y.
• Evaluate how understanding 'one solution' can impact decision-making in applied mathematics or real-life scenarios.
  • Understanding 'one solution' is crucial in applied mathematics because it allows individuals to analyze specific outcomes in various contexts, such as finance or engineering. For instance, knowing that a pricing model yields one solution enables businesses to pinpoint optimal price points for maximizing revenue. Additionally, in fields like project management or resource allocation, recognizing when there is one viable option helps streamline decision-making processes and facilitates efficient problem-solving.

Related terms

Linear Equation: An equation that represents a straight line when graphed, typically in the form $$y = mx + b$$, where m is the slope and b is the y-intercept.

Variable: A symbol used to represent an unknown value in an equation, commonly denoted by letters such as x or y.

Solution Set: The collection of all possible solutions to an equation, which can include one solution, no solutions, or infinitely many solutions.