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

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Key Concepts

Line Relationships

In the context of two-dimensional geometry, the relationship between two lines is determined by comparing their slopes and intercepts. If two lines have the same slope and the same intercept, they are identical; if they have the same slope but different intercepts, they are parallel and do not intersect; and if their slopes differ, the lines intersect at a single point.

Algebraic Solution Methods

Algebraic methods such as substitution or elimination allow for solving systems of equations without graphing. These methods involve manipulating the equations to isolate one variable and substitute it into another equation, thereby reducing the system to a simpler form that reveals the solution type (unique, none, or infinitely many solutions).

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find the values of the variables that satisfy every equation in the system simultaneously. Systems can have a unique solution, infinitely many solutions (if the equations represent the same line), or no solution (if the lines are parallel and distinct).

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