91Ó°ÊÓ

One of the most common ways to express the equation of a line is through the slope-intercept form. This form is written as: \(y = mx + b\). Here, the letter \(m\) represents the slope of the line while \(b\) stands for the y-intercept.

The slope \(m\) indicates how steep the line is and the y-intercept \(b\) shows where the line crosses the y-axis. This way, with the slope-intercept form, you can easily graph the line or understand its steepness and starting point at a glance.

In our exercise, we rewrote the given line equation \(y - x = 13\) in slope-intercept form by solving for \(y\). This helps us easily identify that the slope of the given line is 1, as seen in the equation \(y = x + 13\).

Parallel lines are lines in a plane that never cross each other. They always stay the same distance apart. An essential property of parallel lines is that they have the same slope.

For example, if one line has a slope of 2, a parallel line must also have a slope of 2.

In our exercise, since the original line has a slope of 1, any line parallel to it must also have a slope of 1. This property makes finding equations of parallel lines straightforward once we know the slope.

Calculating the slope of a line helps us understand its direction and steepness. The slope is determined by the rise (change in y) over the run (change in x) between two points on the line. This relationship is mathematically represented as: \(m = \frac{\text{rise}}{\text{run}} = \frac{\text{\text{change in } y}}{\text{change in } x}\).

In slope-intercept form, the slope is the coefficient of \(x\). In our exercise, converting \(y - x = 13\) to \(y = x + 13\) reveals that the slope \(m\) is 1.

To find the equation of a parallel line with a given y-intercept, we used the slope-intercept form and substituted in the slope 1 and y-intercept 0, resulting in \(y = x\).