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Graphing linear equations involves plotting straight lines on a coordinate plane. For each equation, we want to rewrite it in the slope-intercept form, which is given by \( y = mx + b \). This format clearly shows the slope \( m \) and the y-intercept \( b \), which are essential for graphing.

• Slope \( m \): Indicates the steepness of the line, calculated as the change in \( y \) (rise) over the change in \( x \) (run).
• Y-intercept \( b \): The point where the line crosses the y-axis.

For example, take the equation \( y = \frac{x}{2} + \frac{3}{2} \). Here, the slope is \( \frac{1}{2} \) meaning for every 2 units we move horizontally, the line moves 1 unit vertically. The y-intercept is \( \frac{3}{2} \), the starting point on the y-axis.Plotting involves starting at the y-intercept and following the slope to find another point on the line. Repeat this for the second equation \( y = 3x - 1 \), where the slope is 3, and the y-intercept is -1. After plotting both lines, look for where they intersect. This point is the solution of the system if it exists.

Systems of equations are either consistent or inconsistent. This classification depends on whether the system of equations has at least one solution.

• Consistent Systems: These have at least one solution. This means the lines representing the equations on a graph will intersect at some point.
• Inconsistent Systems: These do not have any solutions. On a graph, the lines will be parallel, never meeting each other.

In our example, the lines representing the system intersect at the point (1, 2), confirming that the system is consistent. By definition, any system where the equations' graphical representations meet at any single or multiple points is consistent. In such systems, solutions exist that satisfy both equations.

When dealing with consistent systems, equations can either be dependent or independent. This distinction explains how equations relate to each other graphically.

• Dependent Equations: In these, one equation can be derived from the other. Graphically, it means the equations represent the same line, resulting in infinitely many solutions – every point on the line is a solution.
• Independent Equations: These are distinct and not derived from each other. They intersect at only one point, providing exactly one solution to the system.

In our context, since the system intersects at exactly one point, the equations are independent. Each equation represents a different line, and as these lines cross at a single intersection, they share only that one solution set: (1, 2). Understanding this distinction helps determine how equations relate and the nature of the solutions they provide.