Understanding Systems of Equations with Infinitely Many Solutions

Understanding systems of equations with infinitely many solutions is crucial for various mathematical and real-world applications. These systems represent cases where the equations are interdependent, leading to an infinite number of solutions. Let's explore this concept with examples and explanations.

Dependent Systems of Equations

A common type of system that has infinitely many solutions consists of equations that are interdependent (dependent equations). This means that one equation is simply a multiple or a rearrangement of the other. For instance, consider the system:

Equation 1: (x 2y)

Equation 2: (x 4y)

Here, if we substitute (x 2y) into the second equation, we get (4y 2y). This simplifies to (2y 0), leading to (y 0). Substituting (y 0) back into (x 2y) gives (x 0). However, this is not a unique solution; instead, it implies that the two equations are essentially the same. Any point that satisfies one equation will automatically satisfy the other. Thus, the system has infinitely many solutions that can be represented as the line (x 2y).

General Form of Dependent Systems

A more general form of this system can be written as:

Equation 1: (ax - by 0)

Equation 2: (dx - ey 0)

If these two equations are to have the same solution set, the coefficients must satisfy the condition (ae - bd 0). This condition ensures that one equation is a multiple of the other, leading to a dependent system with infinitely many solutions.

Under-Determined Systems

Example of an Under-Determined System

A system of equations can also have infinitely many solutions when it is under-determined. This occurs when you have fewer equations than unknowns. For example, consider the following system with three unknowns (x, y, z) and two equations:

Equation 1: (ax by cz d)

Equation 2: (px qr y rz s)

Here, (a, b, c, d, p, q, r, s) are constants, and the system has more unknowns than equations. This makes it impossible to find a unique solution for each variable. Instead, you can express one variable in terms of the others.

Solving an Under-Determined System

Let's consider a specific example:

Equation 1: (x 2y 3z 4)

Equation 2: (2x 4y 6z 8)

Notice that the second equation is simply a multiple of the first equation (multiplied by 2). Therefore, the system is equivalent to the single equation:

Equation: (x 2y 3z 4)

From this equation, you can express (x) in terms of (y) and (z), for example, (x 4 - 2y - 3z). This means that for any values of (y) and (z), you can find a corresponding value of (x) that satisfies the equation. Thus, the system has infinitely many solutions.

Conclusion

In conclusion, systems of equations with infinitely many solutions can arise from either dependent equations or under-determined systems. Understanding these concepts is essential for solving complex mathematical problems and has numerous practical applications. Whether through dependent equations or under-determined systems, the key is recognizing the relationships between the equations and the number of variables involved.