Without graphing, decide. See Examples 7 and 8 . a. Are the...

Without graphing, decide. See Examples 7 and $8 .$
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?
$$
\left\{\begin{array}{l}
{3 x+y=1} \\
{3 x+2 y=6}
\end{array}\right.
$$

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Key Concepts

Slope-Intercept Form and Line Comparison

Rewriting linear equations in slope-intercept form (y = mx + b) allows for a clear identification of the slope and y-intercept of each line. Comparing the slopes is particularly useful: if the slopes are different, the lines will intersect at one point; if the slopes are the same but the intercepts differ, the lines are parallel; if both the slopes and intercepts match, the lines coincide.

Types of Solutions in Linear Systems

Systems of linear equations can have one unique solution (where the lines intersect at a single point), infinitely many solutions (when the equations describe the same line, meaning they are identical), or no solution (when the lines are parallel and do not intersect). Recognizing these cases is key to understanding the behavior of the system.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solving such a system requires finding the values of the variables that simultaneously satisfy every equation in the system, and there are multiple methods to do so, including substitution, elimination, and graphical analysis.

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