91Ó°ÊÓ

Linear dependence occurs when one equation in a system can be expressed as a combination of other equations in the same system. In the given exercise, all three equations are multiples of the first equation:
- The second equation is three times the first equation.
- The third equation is five times the first equation.
This means the equations do not provide unique and independent information about the variables. They essentially describe the same relationship among the variables. When equations are linearly dependent, they do not span the entire solution space independently. This characteristic results in either no solutions or infinitely many solutions, depending on whether the equations are consistent.

Having infinite solutions in a system of linear equations means there are an unlimited number of solutions that satisfy all the equations simultaneously. In the exercise, we determined that all equations represent the same plane in a 3D space. Because the planes overlap completely, any point on one plane lies on all three planes. Therefore, the number of solutions isn't limited to a particular point or line but extends infinitely. This happens because any solution to one equation will automatically be a solution to the others, making the system consistent but not independent. When you see this kind of dependency, it's crucial to understand that every variable combination satisfying one equation will satisfy the others.

To describe the infinite solutions in a more concrete way, we use parametric representation. This method involves expressing the variables in terms of one or more parameters. In the given exercise, we can let one variable be a free parameter (let's say, \(z = t\)):
- Start with the first equation: \(2x - 6y + 4t = 8\).
- Simplify to solve for \(x\) and \(y\):
\(x - 3y + 2t = 4\).
- Since \(z = t\), we can express\(x\) and\(y\) in terms of\(t\): \(x = 3y - 2t + 4\). This gives us a parametric form of the solution:

• \(x = 4 + 2t + 3(-y)\)
• \(y = y \)
• \(z = t \)

With this representation, any value of the parameter \(t\) will yield a specific solution \( (x, y, z) \). This approach helps visualize and understand the infinite solutions more clearly and provides a systematic way to list the solutions.