91Ó°ÊÓ

In coordinate geometry, we deal with points on a plane, described by a pair of numbers, known as coordinates. Each point has two coordinates: the first number is the x-coordinate which determines the horizontal position, and the second is the y-coordinate which determines the vertical position.
In a two-dimensional graph, you can think of the x-axis as going left and right, while the y-axis goes up and down. To find the location of any point on this plane, you'll look at the place where these two values intercept.
Points like \(P(2,2)\) and \(Q(-10,0)\) are examples of such coordinates. Here, point \(P\) is situated at 2 steps to the right and 2 steps up from the origin \(0,0\). Point \(Q\), on the other hand, is located 10 steps to the left on the x-axis, staying level on the y-axis.

The slope formula is a fundamental concept in coordinate geometry. It helps us understand the steepness or incline of a line that connects two points on a graph. The slope is an essential characteristic because it directly affects how a line behaves in the plane.
The formula to calculate the slope \(m\) between two points with coordinates \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula calculates the rate at which the line rises vertically per unit of horizontal move. The numerator indicates the change in the y-coordinates, while the denominator shows the change in the x-coordinates between the two points.
For the points \(P(2,2)\) and \(Q(-10,0)\), applying the formula involves substituting these coordinates:
Replace \(y_1\) with 2, \(y_2\) with 0, \(x_1\) with 2, and \(x_2\) with -10.

Linear equations are equations that represent straight lines on a two-dimensional graph. Each linear equation can be expressed in the standard form as \(y = mx + c\), where:

• \(m\) is the slope, describing how steep the line is, and
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

The equation gives a direct relationship between x and y for any point on the line. To find the equation of a line, you need to know either two points on the line, or the slope and the y-intercept.
With a known slope of \(\frac{1}{6}\) from our earlier calculation, the equation for the line through points \(P\) and \(Q\) could be formed if we also know where it crosses the y-axis.
Understanding this equation format makes it easy to predict or calculate values for any point along that line.