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An equation of a line is a mathematical way to represent a straight line in the coordinate plane. It describes all the possible points along that line. When given a specific slope and a point, you can use the point-slope formula to write this equation. The point-slope form is particularly handy when you have one point and the slope of the line. The formula is expressed as: \[ y - y_1 = m(x - x_1) \] Here, \( m \) stands for the slope, and \((x_1, y_1)\) is a known point on the line. Once you substitute the slope and coordinates of the point into this formula, you obtain the line's equation.Using the example with a slope \( m = -\frac{3}{4} \) and the point \((0,0)\), the equation simplifies to \( y = -\frac{3}{4}x \). Whether you start with a point-slope formula or slope-intercept form, the final equation will represent the same line.

The slope-intercept form is a popular and straightforward formula used to express the equation of a line. It is written as: \[ y = mx + b \] In this equation, \( m \) is the slope of the line, and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. When you look at a line's equation in slope-intercept form, it's easy to identify both the slope and the y-intercept at a glance. For instance, in the equation \( y = -\frac{3}{4}x \), it is in slope-intercept form already. Here, the slope \( m = -\frac{3}{4} \), while the y-intercept \( b = 0 \) since there's no added constant (other than zero). This means the line crosses the y-axis at the origin, which is the point \((0,0)\). The slope-intercept form is very efficient for graphing, as it allows you to quickly plot the y-intercept and use the slope to find other points on the line.

The slope of a line is a measure of its steepness and direction. It is calculated as the 'rise' over 'run,' or the change in the vertical position divided by the change in the horizontal position between any two points on the straight line. Mathematically, the slope \( m \) can be represented by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line. A positive slope means the line rises as it moves to the right, while a negative slope indicates it falls. A slope of zero represents a horizontal line, and an undefined slope corresponds to a vertical line.In the given example, the slope \( m = -\frac{3}{4} \) tells us that for every four units the line moves horizontally to the right, it moves three units down. Understanding the slope is key in various applications, from predicting trends to navigating geographic landscapes.