Question

Show that a system consisting of exactly one linear equation can have no solution, one solution, or infinitely many solutions. Give examples.

Short answer

Single linear equations can have no solution, one solution, or infinitely many solutions.

Step-by-step solution

Understanding Concepts

A single linear equation in two variables, for example, has the form \( ax + by = c \). Depending on its parameters, it can represent different geometric entities such as a line or a point in a coordinate plane.

Case Analysis: One Solution

A single equation like \( 2x + 3y = 5 \) generally represents a line. Any specific pair \((x_0, y_0)\) that satisfies the equation is the single intersection point of all possible solutions, thus generating infinite possibilities along that line.

Case Analysis: No Solution

An equation like \( 0x + 0y = 5 \) is impossible because adding zero to zero can never equal a non-zero constant. Thus, this represents no points on a plane, hence no solution exists.

Case Analysis: Infinitely Many Solutions

In the case of an equation like \( 0x + 0y = 0 \), any \((x, y)\) pair satisfies this equation, hence representing the entire coordinate plane. Thus, there are infinite solutions.

Conclusion

When a single linear equation is perfectly satisfied (e.g., \( 0 = 0 \)), it has infinite solutions. If it is never satisfied (e.g., \( 0 = 5 \)), it has no solutions. For other coefficients not leading to trivial contradictions or identities, the equation has exactly one solution as points lying on that line.

Key concepts

No Solution System

A system represented by a single linear equation can sometimes have no possible solutions. This occurs when the equation itself is impossible to satisfy. For instance, consider the equation \(0x + 0y = 5\). Here, the terms on the left side add up to zero, and it's set equal to a non-zero number like 5. Since zero can never equal five, no values of \(x\) and \(y\) will ever make this equation true.

In the realm of linear equations, whenever you find that a constant non-zero value is obtained from the sum of zero coefficients, it translates into the equation having no solutions. Such equations don't correspond to any point or line in geometry; they are essentially void of intersection in a coordinate plane.

One Solution System

In some cases, a single linear equation will have exactly one solution, but more accurately, it can represent a scenario where every pair \((x, y)\) following the equation lies on a line. For example, consider the equation \(2x + 3y = 5\). This equation describes a line in a two-dimensional space. Every point on this line is a solution to the equation.

To visualize what having one solution means, imagine finding the specific point \((x_0, y_0)\) that satisfies the equation perfectly when the context is such that only one intersection point is considered due to specific constraints. However, when not restricted to such settings, the line itself represents infinite solution pairs \((x, y)\) as it stretches across the plane, infinitely satisfying the equation along its length.

Infinite Solutions System

An infinite solution system is essentially a linear equation that holds true for countless pairs \((x, y)\). Take the equation \(0x + 0y = 0\), for example. Since it holds true irrespective of the values of \(x\) and \(y\) (as both sides equal zero), every point on the plane is a solution.

When assessing for infinite solutions, look for situations where the equation simplifies to a tautology, such as \(0 = 0\). This signifies that the line of solutions isn't just a line through specific points but covers the whole plane, highlighting the difference between this scenario and equations representing finite lines or points. Such equations can be visualized as having no boundary on a coordinate system, representing "all-encompassing" solutions.