91Ó°ÊÓ

To find the equation of a line through two points, we start by calculating the slope, often denoted as \( m \). The slope is essentially the measure of steepness or direction of the line. To calculate it, we use the formula:

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

This formula represents the change in the \( y \)-value divided by the change in the \( x \)-value between two points, expressed as \((x_1, y_1)\) and \((x_2, y_2)\).
For example, given points \((-2, 1)\) and \((2, 3)\), we substitute these into our slope formula resulting in:
\[ m = \frac{3 - 1}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}. \]So, the slope \( m \) of the line is \( \frac{1}{2} \), indicating a gentle incline as we move from left to right along the x-axis.

Once we have the slope, we can use the point-slope form to find the equation of the line. The point-slope form is very handy for this because it directly uses a point on the line and the slope:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is any point on the line, and \( m \) is the slope. For our example, we'll use the point \((-2, 1)\) and the slope \( \frac{1}{2} \):
\[ y - 1 = \frac{1}{2}(x + 2). \]By plugging these values into the point-slope form, we establish an equation that represents our line before simplifying it.
Point-slope form is useful because it makes it straightforward to incorporate any known point, making it relatively easy for computations — especially on tests or quizzes where you need to write equations quickly.

Finally, to write the equation in the most commonly used form, we convert it into the slope-intercept form. This form is great because it clearly shows the slope and the y-intercept:

• \( y = mx + b \)

In this equation, \( m \) is the slope, and \( b \) is the y-intercept (where the line crosses the y-axis).
From our point-slope form \( y - 1 = \frac{1}{2}(x + 2) \), we distribute the slope:\[ y - 1 = \frac{1}{2}x + 1. \]Then, we add 1 to both sides to solve for \( y \):
\[ y = \frac{1}{2}x + 2. \]This final equation \( y = \frac{1}{2}x + 2 \) is the slope-intercept form. It shows how the line rises \( \frac{1}{2} \) for every 1 unit moved to the right and intersects the y-axis at 2.