91Ó°ÊÓ

A vertical line is a straight line on the coordinate plane where all points have the same x-coordinate. This means that the line does not tilt to the right or left; it's perfectly up-and-down. The equation that describes a vertical line is of the form \(x = a\), where \(a\) is a constant. This implies that every point on this line has an x-value of \(a\).
In the provided exercise, the equation \(x = 3\) represents a vertical line passing through all points where the x-coordinate remains at 3. Some key features of vertical lines include:

• No slope: Vertical lines cannot be described by a slope (rise/run) as they rise infinitely without running horizontally at all.
• Undefined slope: Mathematically, the slope of a vertical line is undefined because it would require division by zero.
• Intersection with the x-axis: The line crosses the x-axis at (3,0) in this case.

Understanding vertical lines helps us solve intersection problems smoothly by making it evident that any point on the line must share the x-value stated in the equation.

A horizontal line is a line on the coordinate plane where all points have the same y-coordinate, meaning it spans perfectly left-to-right without any vertical incline. The equation of a horizontal line is of the form \(y = b\), where \(b\) remains constant.
In this context, the equation \(y = -2\) represents a horizontal line that crosses the y-axis at (0,-2), maintaining a y-coordinate of -2 across its entire length.A few important characteristics of horizontal lines include:

• Zero slope: The slope of a horizontal line is 0 since there's no rise as the line moves horizontally.
• Parallel to the x-axis: Horizontal lines never intersect the x-axis unless \(y = 0\).
• Uniform y-value: All points along the line share the same y-coordinate.

Recognizing how horizontal lines behave provides clarity when finding intersections, as any point on the line has to match the y-value given in its equation.

To find the point of intersection using algebraic methods, we look for a point that satisfies the conditions of both given line equations. The intersection represents where the two lines meet and coexist on the graph.
In algebraic solutions, the main aim is to solve the equations simultaneously.In the exercise, consider:

• The first line has the equation \(x = 3\), so any intersection point must have an x-coordinate of 3.
• The second line has the equation \(y = -2\), meaning any intersection point must carry a y-coordinate of -2.

By extracting these conditions, we pinpoint the intersection point as (3, -2). This is the unique point where both the vertical and horizontal lines meet. Solving such problems demands understanding how changes in x and y impact each other visually and numerically, consolidating the importance of algebra in geometry.