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To find the slope of a line that passes through two points, we use the slope formula. Slope is a measure of steepness and is essential to defining a line. The formula for slope \( m \) is given by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Essentially, the formula calculates the change in \( y \) (vertical change) over the change in \( x \) (horizontal change).
Think of it as the rise over run. In the given problem, substituting the points \((1, 1)\) and \( \left(6, -\frac{2}{3}\right) \) into the formula, we calculate the slope as follows:

• \( m = \frac{-\frac{2}{3} - 1}{6 - 1} = \frac{-\frac{5}{3}}{5} = -\frac{1}{3} \)

This slope of \(-\frac{1}{3}\) tells us the line falls one unit for every three units it moves to the right.

The slope-intercept form is a commonly used format for linear equations that allows easy access to the slope and y-intercept of a line. It is written as:

• \( y = mx + b \)

Where \( m \) is the slope and \( b \) is the y-intercept. This form is useful because it provides a clear understanding of the line's behavior at a glance.
To use the slope-intercept form, you need the line’s slope and one point on the line to find \( b \).
Using the equation \( y = -\frac{1}{3}x + b \), we substitute the point \((1,1)\):

• \( 1 = -\frac{1}{3}(1) + b \)
• Simplifying gives: \( 1 = -\frac{1}{3} + b \)
• \( b = 1 + \frac{1}{3} = \frac{4}{3} \)

Thus, the y-intercept \( b \) is \( \frac{4}{3} \), meaning the line crosses the y-axis at this point.

A linear equation is an algebraic equation describing a straight line. It involves variables that do not have exponents other than one, making it a first-degree polynomial.
Linear equations can be expressed in various forms, but the slope-intercept form \( y = mx + b \) is often easiest for graphing and analysis.
For a line passing through \((1, 1)\) with a slope of \(-\frac{1}{3}\), and y-intercept \( \frac{4}{3} \), the equation is:

• \( y = -\frac{1}{3}x + \frac{4}{3} \)

This equation shows how \( y \) depends on \( x \), with changes in \( x \) affecting \( y \) according to the slope. Adding the y-intercept adjusts the entire line's position up or down on the graph. Understanding linear equations is crucial not only in mathematics but also in real-world applications like physics and economics, as they model relationships that have a constant rate of change.