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When it comes to writing linear equations, one of the most user-friendly versions is the slope-intercept form, denoted as \(y = mx + c\). In this equation, \(m\) represents the slope of the line, which tells us how steep the line is. The slope is calculated as the rise over the run, meaning the change in \(y\) over the change in \(x\). The \(c\), on the other hand, is the y-intercept, where the line crosses the y-axis. This form is particularly useful because by knowing the slope and the intercept, you can easily graph the line, or by looking at a graph, determine the equation by observing these two characteristics.

In our problem, once we know the slope and a point that the line passes through, we can plug these values into the slope-intercept formula to write the equation of the line. Additionally, the slope-intercept form is ideal for solving various problems involving lines, such as predicting values, finding intercepts, and interpreting the rate of change of a dataset.

Calculating the slope is a fundamental skill when studying linear equations. The slope is determined by taking two distinct points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \) and using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula gives rise to the concept of rise over run, depicting how much the line rises (or falls) for a given horizontal distance (run) between two points.

A positive slope means the line is increasing as it moves from left to right, while a negative slope indicates the line is decreasing. A zero slope means the line is horizontal, and an undefined slope due to a zero denominator corresponds to a vertical line. In our example, we use the coordinates given to calculate the slope. By plugging them into our slope formula, we determine that the line has a steep, negative incline, indicated by a slope of -13.

The equation of a line provides a way to express the relationship between two variables, typically \(x\) and \(y\). There are various forms an equation of a line can take, including slope-intercept, point-slope, and standard form. Each form has its own advantages, depending on what information is provided or what is required for a given math problem.

For this exercise, completing the equation of a line involves using the slope-intercept form and the slope we've previously calculated. With the slope \(m = -13\) and knowing the line passes through the y-axis at \(y = -6\) when \(x = 0\), we get the equation of a line as \(y = -13x - 6\). This equation symbolizes all the points that lie on the line, forming the exact straight path through the points \( (0, -6) \) and \( (-1, 7) \). Understanding how to piece together these components is key to mastering the skill of writing linear equations.