Which Graph Shows A System Of Equations With One Solution

Unveiling the Single Solution: Identifying Systems of Equations with One Unique Answer

Finding the solution to a system of equations is a cornerstone of algebra, with applications spanning numerous fields from engineering and economics to computer science and data analysis. Understanding how to visually identify a system with one unique solution is crucial for both conceptual understanding and efficient problem-solving. This article will delve into the graphical representation of systems of equations, focusing specifically on identifying those systems that yield a single, unique solution. We’ll explore various scenarios, provide clear explanations, and equip you with the tools to confidently determine the number of solutions a system possesses by simply examining its graph.

Understanding Systems of Equations

A system of equations is a collection of two or more equations that contain the same variables. The solution to a system is the set of values for the variables that satisfy all equations simultaneously. Graphically, each equation represents a line (in a two-variable system) or a plane (in a three-variable system). The intersection points of these lines or planes represent the solutions.

We'll primarily focus on two-variable systems (represented by lines) in this article, as the principles extend to higher dimensions. A system of two linear equations can have three possible outcomes:

1. One unique solution: The lines intersect at exactly one point. This is the focus of our discussion.
2. Infinitely many solutions: The lines are coincident (they overlap completely).
3. No solution: The lines are parallel and never intersect.

Graphical Representation of a System with One Solution

A system of equations with one solution is characterized graphically by two lines that intersect at a single point. This point of intersection represents the coordinates (x, y) that satisfy both equations. The x-coordinate is the value of x that solves the system, and the y-coordinate is the corresponding value of y.

Consider the following system of equations:

• Equation 1: y = 2x + 1
• Equation 2: y = -x + 4

To visualize this, imagine plotting both lines on the Cartesian coordinate plane. Equation 1 has a slope of 2 and a y-intercept of 1. Equation 2 has a slope of -1 and a y-intercept of 4. These lines will intersect at a specific point. Finding this point algebraically (through substitution or elimination) will confirm the unique solution. Let’s solve it using substitution:

Since both equations are equal to 'y', we can set them equal to each other:

2x + 1 = -x + 4

Solving for x:

3x = 3 x = 1

Now substitute x = 1 into either equation to find y:

y = 2(1) + 1 = 3

Therefore, the solution to this system is (1, 3). Graphically, the lines intersect at the point (1, 3), demonstrating a system with only one solution.

Identifying One Solution Graphically: Key Features

When examining a graph to determine if a system has one solution, look for these key features:

• Two distinct lines: The graph should show two separate lines, not overlapping or parallel lines.
• Lines intersecting at a single point: The crucial indicator is the presence of only one point of intersection between the two lines. This point represents the unique solution to the system.
• Different slopes: Two lines that intersect must have different slopes. If the slopes were the same, the lines would either be coincident (infinite solutions) or parallel (no solutions).

Analyzing Different Scenarios

Let's explore some examples illustrating systems with one, zero, and infinitely many solutions to reinforce the graphical identification process:

Scenario 1: One Unique Solution (Already Illustrated Above)

The system:

• y = 2x + 1
• y = -x + 4

Graphically, you'll observe two lines intersecting at a single point (1, 3).

Scenario 2: No Solution

The system:

• y = 2x + 1
• y = 2x + 4

Notice that both lines have the same slope (2) but different y-intercepts. This means the lines are parallel and will never intersect. Graphically, you'll see two parallel lines, indicating no solution.

Scenario 3: Infinitely Many Solutions

The system:

• y = 2x + 1
• 2y = 4x + 2

If you simplify the second equation by dividing by 2, you get y = 2x + 1. This is identical to the first equation. Graphically, you will see only one line because both equations represent the same line. Any point on this line satisfies both equations, resulting in infinitely many solutions.

Beyond Linear Equations: Non-linear Systems

While the focus has been on linear equations, the concept of a unique solution extends to non-linear systems. A non-linear system might involve curves (parabolas, circles, etc.). A non-linear system can still have one unique solution if the curves intersect at only one point. For instance, a parabola and a line might intersect at exactly one point, signifying one unique solution.

Algebraic Methods for Verification

While graphical analysis provides a quick visual check, it's crucial to verify your findings using algebraic methods. The most common algebraic techniques are:

• Substitution: Solve one equation for one variable and substitute it into the other equation.
• Elimination: Multiply equations by constants to eliminate a variable when adding the equations together.

Both methods provide an exact solution, confirming the graphical observation.

Frequently Asked Questions (FAQ)

Q1: Can I accurately determine the number of solutions solely from the graph if the lines are very close together?

A1: While close lines might make precise identification challenging, the principle remains the same. If the lines intersect at one point, it's one solution. The closeness doesn't change the fundamental nature of the intersection.

Q2: How do I handle systems with more than two variables?

A2: For systems with three variables, the graphical representation involves planes in three-dimensional space. One unique solution would be represented by the intersection of three planes at a single point. While visualizing this is more complex, the fundamental principle remains: one intersection point means one solution.

Q3: What if my graph isn't perfectly drawn? Will this affect my analysis?

A3: Minor inaccuracies in graphing can slightly alter the perceived intersection point. However, if the lines clearly intersect at one discernible point, it strongly suggests one unique solution. Always verify your graphical observation with an algebraic method for greater accuracy.

Conclusion

Determining whether a system of equations has one unique solution involves understanding the graphical representation of the equations and identifying the intersection points. Two distinct lines intersecting at exactly one point conclusively indicate a system with one solution. While graphical analysis provides a valuable visual aid, always confirm your findings using algebraic techniques for precise and reliable results. Mastering this skill is essential for success in algebra and its numerous applications across various fields. Remember to consider the slopes of the lines: different slopes guarantee intersection, while the same slopes can indicate parallel lines (no solutions) or coincident lines (infinite solutions). This knowledge forms a strong foundation for tackling more complex mathematical problems involving systems of equations.