The following is a system of three equations in only two...

The following is a system of three equations in only two variables. $$\left\{\begin{array}{r} x-y=1 \\ x+y=1 \\ 2 x-y=1 \end{array}\right.$$ (a) Graph the solution of each of these equations. (b) Is there a single point at which all three lines intersect? (c) Is there one ordered pair $(x, y)$ that satisfies all three equations? Why or why not?

The following is a system of three equations in only two variables.
$$\left\{\begin{array}{r}
x-y=1 \\
x+y=1 \\
2 x-y=1
\end{array}\right.$$
(a) Graph the solution of each of these equations.
(b) Is there a single point at which all three lines intersect?
(c) Is there one ordered pair $(x, y)$ that satisfies all three equations? Why or why not?

Key Concepts

Consistency and Inconsistency

Consistency of a system of equations means that at least one set of variable values satisfies all the equations simultaneously, whereas inconsistency indicates that the equations contradict each other such that no common solution exists. When dealing with multiple equations, checking for consistency is crucial to determine if the system has a unique solution, infinite solutions, or no solution at all.

Overdetermined Systems

An overdetermined system has more equations than variables, which can lead to inconsistency. In such cases, even though any two equations might be consistent with one another, the extra equation may impose a condition that cannot be satisfied by the previously determined solution, indicating that no single solution exists for the entire system.

System of Equations

A system of equations is a set of two or more equations with the same set of variables. Solving the system means finding values for the variables that make every equation true. Such systems can be analyzed using methods like substitution, elimination, or graphing, and they can have one solution, infinitely many, or no solution at all.

Graphing Linear Equations

Graphing linear equations involves plotting the lines in a coordinate system based on their slopes and intercepts. When each linear equation is drawn as a line in the plane, it visually represents all possible solutions to that equation. This method is particularly helpful for understanding the relationship between equations and finding points of intersection.

Intersection of Lines

The intersection point of lines represents the common solution (or solutions) to the equations when their graphs cross at a point. In the context of systems of equations, a unique intersection point indicates a single solution, while parallel lines (no intersection) or coincident lines (infinite intersections) indicate different types of systems.

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