Solve each system of linear equations by graphing. See Examples 3 through 6 { x-2 y=-6 -

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Solve each system of linear equations by graphing. See...

Key Concepts

System of Linear Equations

A system of linear equations involves two or more equations that share the same set of variables. The solution to such a system is a set of values for the variables that satisfy every equation in the system simultaneously. When the system is graphed, each equation is represented by a line, and the point of intersection (if it exists) is the solution.

Graphing Method in Solving Systems

The graphing method consists of rewriting the equations—often in slope-intercept form—to make them easy to plot on a coordinate plane. Each equation becomes a line, and the intersection of the lines, if it exists, represents the solution. This method also visually indicates whether the system has a unique solution, infinitely many solutions, or no solution.

Dependent Systems

A dependent system occurs when the equations represent the same line, meaning that all points on the line are solutions to the system. This situation leads to infinitely many solutions. Recognizing a dependent system through graphing is key to understanding that the equations are not independent; rather, one is a multiple or algebraic rearrangement of the other.

Slope-Intercept Form

The slope-intercept form of a line, y = mx + b, is particularly useful in graphing as it clearly shows the slope (m) and y-intercept (b) of the line. Converting equations into this form facilitates an easier and more accurate graphing process, enabling one to quickly identify the characteristics of the line and its behavior on the coordinate plane.

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