which graph shows a system of equations with one solution

Understanding the Basics of Systems of Equations and Their Graphs

A system of equations consists of two or more equations with the same set of variables. The solution to a system is the set of variable values that satisfy all equations simultaneously. When visualized graphically, each equation in the system corresponds to a line (or curve), and the solutions are represented by the points where these lines intersect. The key question is: which graph shows a system of equations with exactly one solution?

What Does It Mean for a System to Have One Solution?

Definition of a Unique Solution

A system of equations has exactly one solution when there is a single point in the coordinate plane that satisfies all the equations simultaneously. Geometrically, this corresponds to the lines intersecting at exactly one point.

Types of Solutions in Systems

Understanding the possible solutions helps clarify what to look for in graphs:

• No Solution: The lines are parallel and distinct, so they never intersect.
• Exactly One Solution: The lines intersect at a single point.
• Infinite Solutions: The lines coincide, meaning they are the same line.

Graphical Characteristics of Systems with One Solution

Lines Intersecting at a Single Point

The hallmark of a system with one solution is two lines that cross exactly once. This intersection point represents the unique solution.

Distinct Slopes

For two lines to intersect at exactly one point, their slopes must be different:

• If the slopes are different, the lines will eventually cross.
• If the slopes are the same but the y-intercepts differ, the lines are parallel and do not intersect.

Visual Clues in Graphs

When examining graphs, look for:

• Two straight lines that cross at a single point.
• No overlapping lines.
• No parallel lines.

Common Forms of Graphs Showing a Single Solution

Standard Form

In the form \(ax + by = c\), the graph of a linear equation is a straight line. For a system:

• Two lines with different slopes.
• Intersection point indicates the unique solution.

Slope-Intercept Form

Expressed as \(y = mx + b\):

• Different slopes (\(m1 \neq m2\)) guarantee a single intersection point.
• Example: \(y = 2x + 3\) and \(y = -x + 1\).

Graphical Features to Identify

• Lines crossing at one point.
• Lines are neither parallel nor coincident.
• The intersection point represents the unique solution.

How to Determine if a Graph Represents a System with One Solution

Step-by-Step Approach

To analyze a given graph:

1. Identify the lines: Confirm that each line represents an equation in the system.
2. Check for intersection points: Locate where the lines cross.
3. Count the intersection points: Ensure there is exactly one point where they meet.
4. Assess the lines' slopes: Visually estimate whether the lines have different slopes; lines with different slopes will intersect at one point.

Common Mistakes to Avoid

• Confusing parallel lines with coincident lines.
• Assuming that lines crossing at more than one point (which doesn’t happen for straight lines) indicates multiple solutions.
• Overlooking the possibility of curved lines (like circles or parabolas) which are not part of linear systems.

Examples of Graphs Showing a System with One Solution

Example 1: Two Lines Crossing at a Single Point

Imagine a graph with:

• Line A: \(y = 2x + 1\)
• Line B: \(y = -x + 4\)

These two lines have different slopes (2 and -1), so they will intersect exactly once at a point. The intersection point can be found algebraically or visually. The graph visually confirms the presence of a single crossing point.

Example 2: Lines in Different Quadrants

Suppose the lines are:

• Line A: \(y = 0.5x - 2\)
• Line B: \(y = -x + 3\)

They have slopes 0.5 and -1, respectively, which are different, and their crossing point indicates the unique solution.

Example 3: Graphs with Different Slopes and Intercepts

Any pair of lines with different slopes will intersect once, such as:

• \(3x - y = 5\)
• \(x + 2y = 4\)

Their graphs will cross at a single point, which is the unique solution.

Visualizing Graphs with No Solution or Infinite Solutions

Parallel Lines (No Solution)

Graphs where the lines are parallel and do not intersect show systems with no solutions. For example:

• \(y = 2x + 3\)
• \(y = 2x - 1\)

These lines have identical slopes but different y-intercepts.

Coincident Lines (Infinite Solutions)

When the two lines are the same, they overlap entirely:

• \(y = -x + 2\)
• \(2y = -2x + 4\) (which simplifies to the same line)

This indicates infinitely many solutions.

Conclusion: Recognizing the Graph with One Solution

In summary, the graph that shows a system of equations with one solution will feature two lines that intersect exactly at one point. The key characteristics include different slopes, no parallelism, and a clear crossing point. Visually, the lines should cross at a single location, confirming that the system has a unique solution. When analyzing graphs, always look for the line intersection points, their relative slopes, and whether the lines are parallel or coincident. Mastering these visual cues will enable you to quickly identify systems with one solution and deepen your understanding of linear systems and their graphical representations.