A student claims that the following system of equations is...

A student claims that the following system of equations is inconsistent, and that there are an infinite number of solutions. Explain why this answer is wrong. Give the correct description of the system.

$$
\begin{aligned}
& y=\frac{1}{4} x+4 \\
& y=0.25 x+4
\end{aligned}
$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solutions to the system are the variable values that satisfy all the equations simultaneously. Understanding the structure of such systems is essential because they can be classified based on the number of solutions present, which in turn depends on the relationships between the equations.

Consistency versus Inconsistency

A system of equations is considered consistent if there is at least one solution that satisfies every equation. In contrast, an inconsistent system has no solutions, typically because the equations describe parallel lines that never intersect. Recognizing the difference between these two states is key in analyzing and solving systems of equations.

Dependent Systems

A dependent system of linear equations is one where the equations are essentially the same; that is, one equation is a multiple of the other. This scenario results in an infinite number of solutions since every solution of one equation is also a solution of the other. Dependent systems are a type of consistent system and should not be confused with inconsistent systems, which have no solution.

Slope-Intercept Form

The slope-intercept form, expressed as y = mx + b where m is the slope and b is the y-intercept, is a standard way to represent linear equations. It is particularly useful for determining whether two lines are parallel, intersecting, or coincident. Identifying that two linear equations have identical slopes and y-intercepts immediately reveals whether they represent the same line, which affects the classification of the system as consistent and dependent.

Please give Ace some feedback