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 .
$\left\{\begin{array}{l}3 y-2 x=3 \\ x+2 y=9\end{array}\right.$

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Key Concepts

Unique, Infinite, and No Solutions

Systems of linear equations can be classified based on the number of solutions they possess: a unique solution occurs when the lines intersect at exactly one point, infinitely many solutions occur when the lines are identical, and no solution occurs when the lines are parallel and never intersect.

Algebraic Methods for Solving Systems

Techniques such as substitution and elimination allow for solving systems of linear equations without graphing. These methods manipulate equations algebraically to reduce the system to a single equation, facilitating the determination of the intersection point or revealing the nature of the system's consistency.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed in a coordinate system. They form the basis for understanding relationships between variables and are typically written in the form Ax + By = C.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with common variables. Solving such systems involves finding the set of variable values that satisfy all equations simultaneously, which can result in a single unique solution, no solution, or infinitely many solutions.

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