Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. If a system of linear equations has two distinct solutions, then it has an infinite number of solutions.

Step-by-Step Solution

Answer

The statement is false. A system of linear equations having two distinct solutions does not necessarily imply it has an infinite number of solutions.

Step 1: Understanding the statement

The statement in question presumes that if a system of linear equations has two distinct solutions, it has an infinite number of solutions. A solution to a system of linear equations is a set of values for the variables that satisfies all the equations simultaneously. A system of linear equations may have a single solution, infinite solutions, or no solution at all.

Step 2: Evaluating the statement

For a system of linear equations, having two distinct solutions implies that there are two different sets of values for the variables that satisfy all the equations simultaneously. However, for a system of linear equations to have infinite solutions, it means the equations represent the same line or plane (in a 3D case), or in general, the same hyperplane in higher dimensions, which means every point on that line or plane is a solution. This can only happen if the equations are multiples of each other.

Step 3: Final judgement

In a system of two or more linear equations, having two distinct solutions does not imply having an infinite number of solutions. Only when the equations are dependent (multiples of each other) can a system have infinite solutions. Therefore, the statement is false.

Key Concepts

Solutions to Linear Equations

When we talk about solutions to linear equations, we are referring to the values that make all the equations in a system true simultaneously. Linear equations form a system where each equation represents a line in a two-dimensional plane or a hyperplane in higher dimensions.

A solution to a system of linear equations is like a meeting point for these lines or hyperplanes. Depending on how these lines are positioned, you can have three scenarios:

• One solution: This happens when all lines intersect at a single point.
• No solution: When at least two lines are parallel, they never meet, indicating no possible solution.
• Infinite solutions: If all lines overlap each other completely, then every point on these lines is a solution.

What's crucial to understand here is that finding the solutions requires us to look at the way the equations interact with one another. Through methods such as graphing, substitution, or elimination, we can determine whether the system is consistent (having at least one solution) or inconsistent (having no solution).

Infinite Solutions

The concept of infinite solutions may sound bewildering at first, but it makes sense when you picture linear equations as lines on a graph. Imagine two lines that lie directly on top of each other; these lines are considered coincident.

For linear equations, having infinite solutions means every point along the line satisfies all equations in the system. This scenario occurs when the equations are not just similar; they are exactly the same when graphed, or they are equivalent variations of each other (like one equation being a multiple of another).

It's like having multiple ways to say the same thing – different words but identical meanings. This characteristic is perhaps one of the most critical aspects when solving systems because it tells you that the equations don't really provide separate conditions, they're just echoing each other.

Dependent Equations

Dependent equations are like dancers moving in perfect harmony; every step is matched. In mathematical terms, these are equations in a system that essentially describe the same line, plane, or hyperplane. This means they are not independent conditions but rather reflections of each other.

If you transform one dependent equation, you can obtain another. For instance, if you multiply or divide an equation by a non-zero number, you're creating a dependent equation. This transformation doesn’t change the solution set—it remains the same.

Understanding whether equations are dependent is pivotal. It helps us determine if we're dealing with a scenario where there could be infinite solutions, essentially making the exercise of finding unique answers futile. It narrows our focus and guides our problem-solving strategy, ensuring that we don't waste time seeking a diverse set of answers when there's inherently only one 'path' that all equations follow.