Exer. 69-70 : The given points were found using empirical...

Exer. $69-70$ : The given points were found using empirical methods. Determine whether they lle on the same line $y=a x+b,$ and if so, find the values of $a$ and $b$
$$\begin{aligned}
&A(-0.22,1.6968), \quad B(-0.12,1.6528)\\
&C(1.3,1.028) \quad D(1.45,0.862)
\end{aligned}$$

Key Concepts

Empirical Methods

Empirical methods involve collecting and analyzing data from observations or experiments rather than relying solely on theoretical reasoning. When points are determined using empirical methods, slight deviations might occur, so techniques such as checking for consistent slopes are essential to assess their alignment accurately.

Linear Equation

A linear equation in the form y = ax + b represents a straight line on the coordinate plane, where 'a' is the slope (indicating the steepness and direction of the line) and 'b' is the y-intercept (the point where the line crosses the y-axis). This concept is fundamental for understanding relationships that can be modeled as straight-line trends.

Slope

The slope of a line is a measure of its steepness, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Consistent slopes between multiple pairs of points indicate that they all lie on the same straight line.

Y-Intercept

The y-intercept is the point on the graph where the line crosses the y-axis, represented by the value 'b' in the equation y = ax + b. It provides a starting value for the line when x is zero, and along with the slope, it completely determines the line.

Collinearity

Collinearity refers to the property of a set of points lying on the same straight line. By verifying that the slope calculated between any pair of points remains constant, one can determine if all given points are collinear.

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