Final answer:
To solve a system of equations by graphing, one must graph and analyze the relationship between the lines involved such as intersecting, parallel, perpendicular or coincident lines. Each scenario has different characteristics and implications for the solution set.
Step-by-step explanation:
The task at hand involves solving the system of equations by graphing different types of lines: intersecting lines parallel lines perpendicular lines, and coincident lines. To address this:
- a) Intersecting lines: Two lines that cross each other at a single point. To solve, graph two lines with different slopes; the solution is where they intersect.
- b) Parallel lines: Two lines that never intersect because they have the same slope but different y-intercepts. There is no solution as they do not meet.
- c) Perpendicular lines: Two lines that intersect at a right angle. The slopes of such lines are negative reciprocals of each other (e.g., m and -1/m).
- d) Coincident lines: Lines that lie exactly on top of each other, indicating an infinite number of solutions since they intersect at every point along the lines.
For linear equations y = mx + b, the 'm' represents the slope, which determines the steepness and direction of the line, while 'b' denotes the y-intercept, the point where the line crosses the y-axis. When graphing, the slope is shown as the rise over run between two points on the line and changes in 'b' shift the line up (larger intercept) or down (smaller intercept) without changing the slope.