Question

Consider the system of linear equations below. Choose nonzero values for \(a, b\), and \(c\) so the system satisfies the given condition. Explain your reasoning. $$ \begin{aligned} &x+y+z=2 \\ &a x+b y+c z=10 \\ &x-2 y+z=4 \end{aligned} $$ a. The system has no solution. b. The system has exactly one solution. c. The system has infinitely many solutions.

Short answer

For these conditions, the system would have:A) No solution if \(a=1\), \(b=1\), \(c=1\), making the second equation parallel to the first one. B) One solution if \(a=1\), \(b=1\), \(c=1\), making the planes intersect at one point. C) Infinitely many solutions if \(a=1\), \(b=-2\), \(c=1\), making all three planes coincide.

Step-by-step solution

- Solution with no solution

In case a system has no solution, the planes represented by the equations are parallel and do not intersect each other. For example, if we make the second equation parallel to one of the other two, there will be no common solution for all three. By making equation (2) parallel to equation (1), we have, \(a=1\), \(b=1\), and \(c=1\), because the coefficients of \(x\), \(y\), and \(z\) in equation (2) must be the same as equation (1) to make them parallel.

- Solution with one solution

In this case, three planes intersect at one point. Keeping \(a\), \(b\), \(c\) as 1, won't lead to parallel planes, hence one common solution for all the three equations exists. Thus, \(a=1\), \(b=1\), and \(c=1\) will make the system to have a unique solution.

- Solution with Infinitely Many Solutions

In this case, all three planes coincide, leading to infinite solutions. Thus, the second equation must be a linear combination of the first and third equation. On closely observing equation (1) and equation (3), it can be noticed that if \(a=1\), \(b=-2\), \(c=1\), equation (2) will be the result of (equation (1) + equation (3)). This leads to infinite solutions.

Key concepts

No Solution in System of Equations

When dealing with systems of linear equations, it is sometimes possible to encounter a situation where no solution exists. This occurs when you have what is known as inconsistent equations. Imagine having two lines on a plane that never intersect; these are parallel lines, and just like parallel lines, a system of equations with no solution is represented by equations that describe parallel planes.

To craft a system with no solution, you want the equations to represent planes that never meet. In the given exercise, we can achieve this by setting the coefficients of the variables in the second equation to be the same as those in the first equation. By choosing values of a = 1, b = 1, and c = 1, the second equation becomes a direct echo of the first, simply scaled by different constants, establishing parallel planes that do not intersect - hence, no solution. This adherence to parallelism ensures that no matter how you move along one plane, you will never cross over to the other, symbolizing the impossibility of solving the system.

Unique Solution in System of Equations

Contrastingly, a system of equations may also yield exactly one solution. We refer to this as a consistent and independent system. For a single unique solution to exist, the planes represented by each equation must intersect at exactly one point in space.

To configure our example system this way, we must adjust the coefficients a, b, and c so that the second equation is not a sheer replica nor a strict linear combination of the other two. This can be demonstrated by setting all of these coefficients to 1. With a = 1, b = 1, and c = 1, each plane intersects at a unique point. This intersection is the single set of values for x, y, and z that satisfies all three equations. It is akin to finding the one point where three distinct but intersecting paths cross each other, capturing the essence of a unique solution perfectly.

Infinitely Many Solutions in System of Equations

Yet another intriguing possibility in a system of linear equations is the occurrence of infinitely many solutions. This particular scenario arises when all equations in the system essentially describe the same plane or when one of them is a combination of the others, leading to coinciding planes.

In our textbook exercise, to achieve infinitely many solutions, we look for coefficients a, b, and c such that the second equation can be seen as a combination of the first and third equations. A keen observation reveals that by taking a = 1, b = -2, and c = 1, the second equation is exactly the sum of equation one and equation three. By the design of our choices, each point on the plane defined by the second equation is also on the plane defined by the first and third equations. It's like having three different maps that all show the same territory; no matter where you go on this 'terrain,' you're adhering to the guidance all three maps provide, emphasizing the concept of infinitely many solutions.