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 .
$\left\{\begin{array}{l}2 x+y=0 \\ 2 y=6-4 x\end{array}\right.$

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Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Systems of linear equations can have a unique solution, infinitely many solutions, or no solution at all, depending on how the lines represented by the equations interact with each other.

Slope-Intercept Form

The slope-intercept form of a linear equation, written as y = mx + b where m is the slope and b is the y-intercept, is a common way to represent a line and quickly determine its steepness and where it crosses the y-axis. Converting equations to this form makes it easier to compare the slopes and intercepts of different lines in a system.

Types of Solutions in Systems of Equations

Systems of linear equations can result in three types of scenarios: the lines may intersect at exactly one point (indicating a unique solution), be parallel with no intersection point (indicating no solution), or coincide completely (indicating infinitely many solutions). Understanding these possibilities is key to analyzing the relationships between the equations without the need for graphing.

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