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 .
$\left\{\begin{array}{l}3 x-y=2 \\ \frac{1}{3} y=-2+3 x\end{array}\right.$

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Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Analyzing such systems allows you to determine the points of intersection, if any, among the lines represented by the equations. These systems are fundamental in exploring how multiple constraints can define unique solutions, infinitely many solutions, or no solution at all.

Comparison of Linear Equations

Comparing linear equations involves examining their slopes and intercepts. If the slopes are equal, the lines are either parallel (if the intercepts differ) or identical (if the intercepts are also equal). This comparison is essential for determining the relationship between two lines without graphical plotting.

Solution Sets of Linear Systems

The solution set for a system of linear equations represents the set of all points satisfying every equation simultaneously. Depending on the relationship between the equations, the system can yield a unique solution (if the lines intersect at exactly one point), infinitely many solutions (if the lines are identical), or no solution (if the lines are parallel). This concept is central to understanding the behavior and outcome of any system of linear equations.

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