Solving Simultaneous Equations Involving Infinite Solutions

In mathematics, the solutions to systems of equations can range from a single point to a continuum of solutions. This article explores the intriguing scenario where a system of two equations with three unknowns results in an infinite number of solutions. We will delve into the step-by-step method to find these solutions and discuss the implications of such a system.

Introduction to Simultaneous Equations

Simultaneous equations are a set of equations that share the same variables. For example, consider the equations:

Together, these equations form a system that we will solve to determine the possible values of (a), (b), and (c).

Step-by-Step Solution Using Substitution

To solve this system, we can use the method of substitution. Here’s how:

Step 1: Solve Equation 1 for (c)

From Equation 1, we can express (c) in terms of (a) and (b):

Equation 1: (abc 0) (1)

This condition can be satisfied if either (a 0), (b 0), or (c 0). However, for our substitution method, let’s assume (c) needs to be expressed based on (a) and (b):

(c -a - b)

Step 2: Substitute (c) in Equation 2

Now substitute the expression for (c) into Equation 2:

Equation 2: (3a 2b c 0) (2)

Substituting (c -a - b):

(3a 2b - a - b 0)

Combine like terms:

(2a b 0)

Step 3: Solve for (b)

From the equation (2a b 0), express (b) in terms of (a):

(b -2a)

Step 4: Substitute (b) back to find (c)

Now substitute (b -2a) back into the equation for (c):

(c -a - -2a -a 2a a)

Summary of Solutions

At this point, we have:

(b -2a)
and
(c a)

We can express the solutions in terms of (a):

(a, b, c a, -2a, a)
for any real number (a).

This means that the system has infinitely many solutions depending on the value of (a).

The Concept of Infinite Solutions

The fact that the system of equations allows for an infinite number of solutions might be surprising. In a linear algebra context, this indicates that the system is underdetermined. This means that there are more variables than independent equations to solve for them precisely. In such scenarios, the solution set is a line or a higher-dimensional subspace.

Real-World Analogy

Imagine a math problem presented from a textbook:

“Okay class, open your textbooks to page 160 and do questions 3 to 16.”

The problem at hand in question 16 is:

“The equations are (abc 0) and (3a 2b c 0).”

After some thought, a student named Jonathan realizes that:

“It seems there is a problem with the question. There are only two equations for three variables. It cannot be solved.”

The teacher confirms Jonathan’s solution, but then introduces the concept of infinite solutions. The problem of three cars, where the price of each car is a variable, and the relationships between them are equations, is used to illustrate the point:

“The relationships between the price of each car are equations. When I gave you two relations, you couldn’t solve it. When I added the third one, you could figure out the answer. You will find that this applies to any system of however many variables or equations.”

Understanding these concepts is crucial for anyone studying algebra or linear systems, as it helps to recognize when a system is underdetermined and how to work with such systems.

Conclusion: The system of equations (abc 0) and (3a 2b c 0) results in an infinite number of solutions, which can be expressed as (a, b, c a, -2a, a). This exemplifies the concept of underdetermined systems and the importance of recognizing when a subset of variables can take any value, leading to an infinite solution set.