Recall from Chapter 1 that a unique line is determined by...

Recall from Chapter 1 that a unique line is determined by two distinct points on the line and that the values of $m$ and $b$ can then be determined for the general form of the linear function
$$
f(x)=m x+b
$$
Find the equation of this new line, and write it in the form $y_{2}=m x+b$

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Key Concepts

Point-Slope Form

The point-slope form of a linear equation provides a method to construct the line's equation using a known point and the slope, typically written as y - y1 = m (x - x1). This form is especially handy when directly computing the equation from two given points or when translating between different forms of linear equations.

Slope-Intercept Form

The slope-intercept form of a linear equation, expressed as y = m x + b, is a standard representation where m denotes the slope and b represents the y-intercept. This format is useful for easily identifying the rate of change and the point at which the line crosses the y-axis.

Unique Line Determined by Two Points

In Euclidean geometry, any two distinct points determine a unique line. This principle underpins the concept that, given two points, there exists exactly one straight line that passes through both, enabling us to derive its equation without ambiguity.

Slope (Gradient)

The slope of a line is a measure of its steepness and direction, calculated as the ratio of the vertical change to the horizontal change between two distinct points on the line. It is a fundamental attribute in defining the behavior and orientation of a linear function.

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