Problem 3 - Precalculus: Mathematics for Calculus

Question

A system of two linear equations in two variables can have one solution, _____ solution, or _____ _____ solutions.

Step-by-Step Solution

Answer

A system can have one solution, no solution, or infinitely many solutions.

Step 1: Introduction to the Types of Solutions

A system of two linear equations in two variables can have three types of solutions: one solution, no solution, or infinitely many solutions. These outcomes are based on the relationship between the lines represented by the equations.

Step 1: Identify One Solution

When the two lines represented by the equations intersect at a single point, the system has one solution. This means the equations are independent and the coefficients of the variables do not make them parallel.

Step 2: Identify No Solution

If the two lines are parallel and do not intersect, the system has no solutions. This occurs when the lines have the same slope but different y-intercepts, making them distinct parallel lines.

Step 3: Identify Infinitely Many Solutions

When the two lines coincide completely, they represent the same line. In this case, every point on the line satisfies both equations, resulting in infinitely many solutions. This happens when both the slope and y-intercept are identical for the two equations.

Key Concepts

Types of Solutions

In the world of linear equations, understanding the different types of solutions is key to grasping the overall behavior of systems. A "system of linear equations" can yield three distinct types of solutions:

• One Solution: This occurs when two lines intersect at a single point. This single point represents the only solution to the system, indicating that the lines are neither parallel nor coincident.
• No Solution: Sometimes, the equations will describe parallel lines that never meet. In this case, there is no shared point, thus no solution exists.
• Infinitely Many Solutions: When the lines are identical or coincide perfectly, any point on the line is a solution. Thus, there are countless solutions.

Each type of solution provides insights about the geometric relationship between the lines in question.

Parallel Lines

Parallel lines in the plane are like train tracks that never meet. For a system of linear equations, this typically indicates the "no solution" scenario. These lines maintain the same slope—meaning they ascend or descend at the same angle—but have differing y-intercepts, which keeps them separate.

• Example: Consider the equations \( y = 2x + 3 \) and \( y = 2x - 4 \). Both lines have a slope of 2 but different y-intercepts (3 and -4).
• Result: The lines do not intersect, demonstrating they are parallel and thus have no solution in common.

Parallel lines exemplify how two equations can exist side by side without sharing any points in common.

Intersection of Lines

The intersection of two lines on a graph signifies a shared point where both equations hold true. When lines intersect, the coordinates at the point of intersection satisfy both equations, representing the solution to the system.

• Geometric Interpretation: The point of intersection is where two linear paths cross each other's tracks.
• Algebraic Perspective: Finding this intersection involves solving the equations simultaneously, often by methods like substitution or elimination.
• Unique Solution: This crossing point signifies the only solution when the two lines are neither parallel nor identical.

Understanding line intersections helps reveal solutions that arise when distinct paths meet in geometry.