91Ó°ÊÓ

Coordinate geometry, also known as analytic geometry, involves plotting points, lines, and curves on a coordinate plane using algebraic equations. The plane consists of two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical). Each point is represented as a pair \(x, y\), identifying its position.
To find where two lines intersect, one needs to determine the point that satisfies both line equations simultaneously. This point will have coordinates that work for both lines, essentially where the two lines "meet" or cross each other on the plane.
For instance, if a line is described by \(x + 2y = 12\), it defines one relation between x and y. Familiarizing yourself with how linear equations translate to lines within this system can greatly aid in understanding intersections.

Linear equations are polynomials of degree one, and they graph as straight lines on the coordinate plane. Each equation can often be rewritten in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This form makes it easy to identify how steep the line is and where it crosses the y-axis. For example, the equation \(y = 3x - 7\) tells us the slope is 3, and the line crosses the y-axis at \(-7\).

• Parallel lines: Have the same slope but different y-intercepts. They never meet.
• Intersecting lines: Have different slopes and intersect at one point.
• Coincident lines: Are essentially the same line, having identical slopes and y-intercepts.

Knowing how to convert and interpret these equations helps in solving problems related to intersections.

A system of equations consists of multiple linear equations considered together. The solution to a system is the set of points that satisfies all the equations simultaneously.
There are various methods to solve these systems:

• Substitution: Solve one equation for one variable, then substitute that solution into the other equation.
• Elimination: Add or subtract equations to eliminate one variable, solving for another.
• Graphical Method: Graph each equation and find the point(s) where they intersect.

In the given task, substituting \(y = 0\) into the first equation simplifies finding the intersection with the x-axis. In the second task, converting equations to a common form \(y = mx + b\) allowed direct comparison.
Mastering these methods is crucial for tackling complex problems in coordinate geometry.