Without graphing, decide. a. Are the graphs of the equations...

Without graphing, decide.
a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point?
b. How many solutions does the system have? See Examples 7 and 8 .
$\left\{\begin{array}{l}6 y+4 x=6 \\ 3 y-3=-2 x\end{array}\right.$

Key Concepts

Consistency and Types of Solutions

The concepts of consistency and the number of solutions in linear systems involve determining whether a system has a single unique solution (consistent and independent), infinitely many solutions (consistent and dependent), or no solution (inconsistent). These classifications arise from the geometric relationships between lines in the system, such as intersection at one point, coinciding, or being parallel.

Parallel and Coincident Lines

In the context of linear systems, parallel lines are those with equal slopes and different y-intercepts, leading to no point of intersection (no solution). Identical or coincident lines have exactly the same slopes and y-intercepts, meaning that every point on one line is also on the other, resulting in infinitely many solutions. Recognizing these conditions is integral to analyzing the behavior of the system without needing to graph it.

Slope-Intercept Form

Rewriting a linear equation in the slope-intercept form (y = mx + b) allows one to identify the slope and y-intercept directly. This form is essential for quickly determining the relationship between lines, such as whether they are parallel, intersecting, or identical. It helps compare the slopes and intercepts of different equations without graphing.

Linear System of Equations

A linear system consists of two or more linear equations involving the same set of variables. Understanding these systems involves knowing how the lines represented by these equations behave relative to each other, which in turn determines the number of solutions. This subject is fundamental because it includes solving for intersections that represent common solutions to all equations in the system.