State the conditions on the constants a, b, c, and d for...

State the conditions on the constants $a, b, c$, and $d$ for which the system below is (a) consistent and independent, (b) consistent and dependent, and (c) inconsistent.

$$
\begin{aligned}
& y=a x+b \\
& y=c x+d
\end{aligned}
$$

Key Concepts

Linear Equations in Two Variables

These are equations that involve two variables and represent straight lines on a coordinate plane. Their solutions are the points where the lines intersect, and analyzing their relationships helps determine whether a given system has one solution, infinitely many solutions, or no solution at all.

Consistent Systems

A system of equations is consistent if there is at least one solution. In the context of linear equations, consistency means that the lines either intersect at a single point or coincide entirely, providing infinitely many solutions.

Independent Systems

An independent system of linear equations has exactly one unique solution. This occurs when the lines have different slopes, ensuring they intersect at precisely one point. The condition for this in the standard line equations is that the slopes are unequal.

Dependent Systems

A dependent system has infinitely many solutions because the equations represent the same line. For linear equations, this dependency occurs when both the slopes and y-intercepts are identical, meaning one equation is a direct scalar multiple of the other.

Inconsistent Systems

An inconsistent system has no solution, which happens when the lines are parallel and distinct. In linear equations, this means the slopes are identical while the y-intercepts differ, leading the lines to never intersect.

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