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 .
$\left\{\begin{array}{l}4 x-y=6 \\ \frac{1}{2} y=-3+2 x\end{array}\right.$

Key Concepts

Types of Solutions in Linear Systems

In systems of linear equations, the type of solution is determined by the relationship between the lines represented by the equations. A unique solution exists if the lines intersect at a single point. There are infinitely many solutions if the lines are identical (coincident), and no solutions exist if the lines are parallel but distinct.

Slope and Line Comparison

Comparing the slopes of two lines is essential to decide their relationship. If two lines have the same slope and the same y-intercept, they are identical (coincident). If they have the same slope but different y-intercepts, they are parallel and do not intersect. Otherwise, if the slopes differ, the lines will intersect at a single point.

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. The solution to the system can be one unique solution, infinitely many solutions (if the lines are identical), or no solution (if the lines are parallel with different intercepts).

Slope-Intercept Form

The slope-intercept form of a linear equation, given as y = mx + b, is useful for identifying the behavior of the line, where m represents the slope and b the y-intercept. Converting equations to this form makes it easier to compare their slopes and y-intercepts, a key step in determining whether lines are parallel, identical, or intersecting.

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