91Ó°ÊÓ

Graphing linear equations is a crucial skill in understanding linear systems, and it involves representing equations visually on a coordinate plane. When graphing a linear equation, we focus on two primary features: the slope and the y-intercept. Each equation of a line can have its own unique slope and intercept, dictating the line's position and angle on the graph.
To begin the graphing process, at least two points are needed. However, using the slope-intercept form of an equation makes this simple because it gives us a starting point (the y-intercept) and a way to find other points (the slope). Plotting the y-intercept on the y-axis gives the first point, and utilizing the slope guides us to the next. Drawing a straight line through these points completes the graph.
This visual representation helps us see how each line relates to others. It allows the determination of intersections between lines, which is essential for solving linear systems.

The slope-intercept form of a linear equation is highly advantageous for graphing and understanding relationships between variables. Expressed as \( y = mx + b \), it clearly shows the slope \( m \) and the y-intercept \( b \).

• **Slope (\( m \))**: Indicates the steepness of the line, calculated as the vertical change (rise) over the horizontal change (run). It dictates how sharply the line ascends or descends.
• **Y-Intercept (\( b \))**: Represents the point where the line cuts through the y-axis. This is where the value of \( x \) is zero.

In the exercise, transforming the original equations into the slope-intercept form made them easier to graph. For the first equation \( x - y = 4 \), rearranging gives \( y = x - 4 \), revealing a slope of 1 and a y-intercept of -4. For the second \( 2x + y = 2 \), it becomes \( y = -2x + 2 \), with a slope of -2 and y-intercept of 2. Understanding these components creates a straightforward basis for plotting the graph.

The intersection point of two lines on a graph represents the set of coordinates \((x, y)\) that satisfies both equations in the system simultaneously. Identifying this point is the primary goal when solving linear systems through graphing.
When two lines intersect, they meet at a common point where their equations yield the same value for \( y \) given a particular value for \( x \). For the current system of equations, graphing revealed that the lines intersected at the point \((2, -2)\). This tells us that \( x = 2 \) and \( y = -2 \) simultaneously satisfy both equations.
Verifying this intersection point by substituting \( x = 2 \) and \( y = -2 \) back into both original equations showed that the calculations held true, confirming that the graph's intersection indeed provided a valid solution.

The solution of a linear system refers to the values of \( x \) and \( y \) that satisfy all equations involved. For a system of two equations, the solution is the intersection point on a graph, determined visually.
There are three types of solutions a linear system can have:

• **One Solution**: The lines intersect at exactly one point, as in our example at \((2, -2)\), signifying a unique solution.
• **No Solution**: The lines are parallel and do not intersect, implying no shared solutions.
• **Infinitely Many Solutions**: The lines overlap completely, indicating that every point on the lines is a solution.

Graphing each equation helps clearly visualize these possibilities. In practice, confirming the graphical solution by substituting the intersection’s coordinates into the original equations ensures the system is solved accurately.