Which Graph Shows A System Of Equations With One Solution

In the world of algebra, solving systems of equations is a fundamental concept with wide-ranging applications. Understanding the graphical representation of these systems is key to visualizing their solutions. Let's dive deep into how a graph reveals a system of equations with one solution, exploring the underlying principles, different scenarios, and practical examples.

Introduction to Systems of Equations

A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values that satisfy all the equations simultaneously. Graphically, each equation in the system represents a line (for linear equations) or a curve (for non-linear equations) on a coordinate plane. The solution(s) to the system are the point(s) where these lines or curves intersect.

When dealing with linear equations, there are three possible scenarios:

• One Solution: The lines intersect at a single point.
• No Solution: The lines are parallel and never intersect.
• Infinite Solutions: The lines are coincident, meaning they overlap completely.

Our focus here is on the first scenario: a system of equations with one unique solution.

Graphical Representation of a System with One Solution

A system of equations has one solution when the graphs of the equations intersect at exactly one point. This intersection point represents the unique (x, y) values that satisfy both equations. Let's break down the characteristics of such a graph:

Intersecting Lines

The most straightforward example is when the system consists of two linear equations. In this case, each equation represents a straight line. For the system to have one solution, the two lines must intersect at a single point. This means the lines cannot be parallel or coincident.

Non-Parallel Lines

Parallel lines have the same slope but different y-intercepts. They never intersect, indicating no solution. To have one solution, the lines must have different slopes. The point where they cross each other is the unique solution to the system.

Non-Coincident Lines

Coincident lines are essentially the same line represented in different forms. They have the same slope and the same y-intercept. Every point on one line is also on the other line, leading to infinite solutions. For a single solution, the lines must be distinct and intersect at only one point.

Example of a System with One Solution

Consider the following system of linear equations:

1. y = x + 1
2. y = -x + 3

To find the solution graphically, we plot these two lines on the same coordinate plane.

• Line 1: y = x + 1 has a slope of 1 and a y-intercept of 1.
• Line 2: y = -x + 3 has a slope of -1 and a y-intercept of 3.

When plotted, these lines intersect at the point (1, 2). This means x = 1 and y = 2 satisfy both equations. Therefore, the system has one solution: (1, 2).

How to Determine the Number of Solutions from a Graph

Determining whether a system of equations has one solution, no solution, or infinite solutions involves analyzing the graphical representation of the equations. Here’s how you can do it:

Step-by-Step Analysis

1. Plot the Equations: Graph each equation in the system on the same coordinate plane. Ensure the equations are accurately represented.
2. Identify Intersection Points: Look for points where the graphs intersect.
   • If the graphs intersect at exactly one point, the system has one solution.
   • If the graphs do not intersect (i.e., they are parallel), the system has no solution.
   • If the graphs overlap completely (i.e., they are coincident), the system has infinite solutions.
3. Determine the Coordinates: If there is an intersection point, find its coordinates (x, y). These coordinates represent the solution to the system.
4. Verify the Solution: Substitute the x and y values into both original equations to ensure they satisfy both.

Common Scenarios

• Linear Equations: Two straight lines can intersect at one point, be parallel (no intersection), or be the same line (infinite intersections).
• Non-Linear Equations: Curves like parabolas, circles, and hyperbolas can intersect at multiple points, one point, or no points, depending on their equations and positions.

Examples of Different Scenarios

• One Solution:
  • System: y = x, y = -x + 2
  • Graphical Representation: Two lines intersecting at (1, 1)
• No Solution:
  • System: y = 2x + 1, y = 2x - 1
  • Graphical Representation: Two parallel lines
• Infinite Solutions:
  • System: y = x + 1, 2y = 2x + 2
  • Graphical Representation: Two coincident lines

Algebraic Methods to Confirm Graphical Solutions

While graphs provide a visual understanding of the solutions to a system of equations, algebraic methods offer a precise way to find and confirm these solutions. Here are common algebraic methods:

Substitution Method

1. Solve for One Variable: Choose one equation and solve it for one variable in terms of the other.
2. Substitute: Substitute the expression from step 1 into the other equation.
3. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
4. Back-Substitute: Substitute the value found in step 3 back into one of the original equations to find the value of the other variable.
5. Check the Solution: Verify that the solution satisfies both original equations.

Elimination Method

1. Align Equations: Write the equations so that like terms are aligned in columns.
2. Multiply Equations: Multiply one or both equations by a constant so that the coefficients of one variable are opposites.
3. Add Equations: Add the equations together, eliminating one variable.
4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
5. Back-Substitute: Substitute the value found in step 4 back into one of the original equations to find the value of the other variable.
6. Check the Solution: Verify that the solution satisfies both original equations.

Comparison Method

1. Isolate the Same Variable: Choose a variable and isolate it in both equations.
2. Set Expressions Equal: Set the two expressions equal to each other.
3. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
4. Back-Substitute: Substitute the value found in step 3 back into one of the original equations to find the value of the other variable.
5. Check the Solution: Verify that the solution satisfies both original equations.

Examples Combining Graphical and Algebraic Methods

Example 1:

• System: y = 2x - 1, y = -x + 5

Graphical Method:

• Plot the lines y = 2x - 1 and y = -x + 5 on the same graph.
• The lines intersect at the point (2, 3).

Algebraic Method (Substitution):

• Since y = 2x - 1 and y = -x + 5, set the expressions equal to each other:
  • 2x - 1 = -x + 5
• Solve for x:
  • 3x = 6
  • x = 2
• Substitute x = 2 into y = 2x - 1:
  • y = 2(2) - 1
  • y = 3
• The solution is (2, 3), which confirms the graphical solution.

Example 2:

• System: x + y = 4, 2x - y = 2

Graphical Method:

• Rewrite the equations as y = -x + 4 and y = 2x - 2.
• Plot these lines on the same graph.
• The lines intersect at the point (2, 2).

Algebraic Method (Elimination):

• Add the two original equations:
  • (x + y) + (2x - y) = 4 + 2
  • 3x = 6
  • x = 2
• Substitute x = 2 into x + y = 4:
  • 2 + y = 4
  • y = 2
• The solution is (2, 2), which confirms the graphical solution.

Advanced Scenarios and Non-Linear Equations

While linear equations provide a foundational understanding, systems of equations can involve non-linear equations, leading to more complex graphical representations and solutions.

Non-Linear Systems

A non-linear system involves at least one equation that is not linear. Examples include quadratic equations, exponential equations, and trigonometric equations. These equations produce curves (parabolas, circles, etc.) when graphed.

Multiple Solutions

Non-linear systems can have multiple solutions. For instance, a line and a parabola can intersect at two points, one point, or no points. The number of solutions depends on the specific equations and their graphical representations.

Example of a Non-Linear System

• System: y = x^2 (parabola), y = x + 2 (line)

Graphical Method:

• Plot the parabola y = x^2 and the line y = x + 2 on the same graph.
• The graphs intersect at two points: (-1, 1) and (2, 4).

Algebraic Method (Substitution):

• Set the equations equal to each other:
  • x^2 = x + 2
• Rearrange the equation:
  • x^2 - x - 2 = 0
• Factor the quadratic equation:
  • (x - 2)(x + 1) = 0
• Solve for x:
  • x = 2 or x = -1
• Substitute x = 2 into y = x + 2:
  • y = 2 + 2 = 4
• Substitute x = -1 into y = x + 2:
  • y = -1 + 2 = 1
• The solutions are (-1, 1) and (2, 4), which confirms the graphical solution.

Tangent Points

In some non-linear systems, a curve and a line might be tangent to each other, meaning they touch at only one point without crossing. This tangent point represents a single solution to the system.

Practical Applications of Systems of Equations

Systems of equations are not just abstract mathematical concepts; they have numerous practical applications in various fields.

Physics

In physics, systems of equations are used to solve problems involving motion, forces, and energy. For example, determining the trajectory of a projectile often involves solving a system of equations that describe its horizontal and vertical motion.

Engineering

Engineers use systems of equations to design structures, analyze circuits, and model fluid dynamics. These equations help them optimize designs and ensure the safety and efficiency of engineered systems.

Economics

Economists use systems of equations to model supply and demand, analyze market equilibrium, and forecast economic trends. These models help understand complex economic interactions and make informed policy decisions.

Computer Science

In computer science, systems of equations are used in optimization algorithms, computer graphics, and data analysis. For example, solving systems of linear equations is fundamental to machine learning and artificial intelligence.

Real-World Examples

• Mixing Problems: Determining the amounts of different ingredients needed to create a mixture with specific properties often involves solving a system of equations.
• Distance-Rate-Time Problems: Calculating the speeds and travel times of different vehicles based on their relative distances involves setting up and solving a system of equations.
• Investment Problems: Allocating investments among different assets to achieve a desired return while managing risk can be modeled and solved using systems of equations.

Common Mistakes and How to Avoid Them

When working with systems of equations, several common mistakes can lead to incorrect solutions. Here’s how to avoid them:

Misinterpreting Graphs

• Mistake: Failing to accurately plot the equations or misreading the intersection points.
• Solution: Use graph paper or graphing software to plot equations accurately. Double-check the coordinates of intersection points.

Algebraic Errors

• Mistake: Making errors in algebraic manipulations, such as incorrect substitutions or sign errors.
• Solution: Double-check each step in the algebraic process. Use a calculator to verify numerical calculations.

Incorrectly Identifying Solution Types

• Mistake: Confusing systems with no solution (parallel lines) or infinite solutions (coincident lines) with systems having one solution.
• Solution: Carefully analyze the slopes and y-intercepts of the lines. Remember that parallel lines have the same slope but different y-intercepts, while coincident lines have the same slope and y-intercept.

Forgetting to Check Solutions

• Mistake: Not verifying that the solution satisfies both original equations.
• Solution: Always substitute the x and y values back into the original equations to ensure they hold true.

Best Practices for Solving Systems of Equations

To ensure accuracy and efficiency when solving systems of equations, follow these best practices:

Organize Your Work

• Keep your work neat and organized. Use clear notation and label each step in the solution process.

Use Graphing Tools

• Utilize graphing calculators or software to visualize equations and identify potential solutions. These tools can help you quickly check your work and identify errors.

Practice Regularly

• Practice solving a variety of systems of equations to improve your skills and build confidence. The more you practice, the better you will become at recognizing patterns and avoiding common mistakes.

Understand the Concepts

• Focus on understanding the underlying concepts rather than just memorizing formulas. A solid understanding of the principles behind systems of equations will help you solve more complex problems and apply these concepts in real-world scenarios.

Check Your Answers

• Always verify your solutions using both graphical and algebraic methods. This will help you catch errors and ensure that your answers are accurate.

Conclusion

Understanding how a graph shows a system of equations with one solution is a fundamental skill in algebra. By recognizing intersecting lines, identifying intersection points, and verifying solutions algebraically, you can confidently solve a wide range of problems. Whether you're dealing with linear or non-linear systems, the combination of graphical and algebraic methods provides a powerful approach to finding accurate and meaningful solutions.