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 .
$\left\{\begin{array}{l}4 x+y=24 \\ x+2 y=2\end{array}\right.$

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Key Concepts

Types of Solutions in Linear Systems

Linear systems can have three types of solutions: a unique solution (where the lines intersect at one point), infinitely many solutions (when the lines are identical), or no solution (when the lines are parallel and do not meet).

Methods for Solving Linear Systems

Techniques such as substitution and elimination are commonly used to solve systems of linear equations algebraically without needing to graph the equations. These methods simplify the system by reducing the number of variables step-by-step until the solution can be identified.

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 these variables that simultaneously satisfy all the equations in the system.

Graphical Interpretation of Linear Systems

When graphed, each equation in a linear system represents a line on the coordinate plane. The nature of the solution to the system—whether unique, infinite, or none—corresponds to how these lines intersect. For example, if they intersect at a single point, there is a unique solution; if they coincide, there are infinitely many solutions; and if they are parallel, there is no solution.

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