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The slope formula is a key concept in coordinate geometry that helps us understand the steepness or incline of a line between two points. To find the slope between any two points, \(x_1, y_1\) and \(x_2, y_2\), you can use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here’s a breakdown of what each part means:

• \(y_2\) and \(y_1\) are the y-coordinates of the two points.
• \(x_2\) and \(x_1\) are the x-coordinates.

When using the slope formula, you are essentially finding the change in y-values divided by the change in x-values. This ratio gives you the rate at which the y-coordinate changes with respect to the x-coordinate. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. Understanding slope is crucial when determining if multiple points lie on the same straight line, known as collinearity.

When we say points are collinear, it means they lie on the same straight line. For three points to be collinear, the slope between any two pairs of points must be the same. Imagine three points: A, B, and C. To prove they are collinear, you could:

• Calculate the slope between points A and B.
• Calculate the slope between points A and C.
• Ensure these two slopes are equal.

In this exercise, to determine if the points \((-6, 2)\), \((-5, y)\), and \((-3, 5)\) lie on a single line, we:

• Found the slope \(m_1\) between the first and third points: 1
• Found the slope \(m_2\) between the first and second points: \(\frac{y - 2}{1}\)

By setting these slopes equal, \(m_1 = m_2\), we demonstrated that the unknown y-value can be solved to ensure that all points align on the same line.

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system allows us to represent geometric shapes, like lines and curves, using algebraic equations. By plotting points on the coordinate plane, we can analyze relationships and calculate key attributes like slope and distance. In this problem, coordinate geometry helps us explore the concept of collinearity by transforming geometric relationships into algebraic equations. We use:

• The coordinate plane to plot our points.
• Algebra to solve for unknowns like y.

This branch of mathematics combines pure geometry with algebra to empower us with tools to solve complex problems in a structured, logical way. As we connect mathematical concepts in the context of real-world coordinates, it deepens our understanding and provides a practical application for theoretical knowledge.