91Ó°ÊÓ

The slope of a line is a measure of its steepness. If you want to find the slope of a line that passes through two points, you use the slope formula. This helpful formula is written as:\[(m = \frac{y_2 - y_1}{x_2 - x_1})\]
The variables \(m\) denote the slope of the line, while \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.
Using the example given with the points \((2,3)\) and \((6,-3)\), you can easily find the slope:

• First, substitute the coordinates into the formula like this: \((-3 - 3)/(6 - 2)\)
• Simplify the expression to get \(-6/4\)
• Finally, reduce to find the slope \(m = -1.5\)

With this value of slope, you now understand the relationship between the points and how steep they are in relation to one another. The slope tells you that for every unit you move to the right (a positive increment in the x-direction), the y-value decreases by 1.5 units.

The y-intercept is a crucial part of understanding the full equation of a line. It is simply the point where the line crosses the y-axis. To find the y-intercept when you know the slope, you use the formula:
\[y - mx = c\]
Here, \(c\) represents the y-intercept, \(m\) is the slope, and \(x\) and \(y\) are the coordinates of a point on the line.
From the earlier example, we know the slope \(m = -1.5\) and we can use one of the points on the line, such as \((2,3)\). Substituting these values into the formula gives us:

• First, write the expression: \(3 - (-1.5 \times 2) = c\)
• Simplify it to \(3 + 3 = c\)
• After simplification, we find \(c = 6\)

Thus, the y-intercept is at the point \((0, 6)\). This means the line crosses the y-axis at the value of 6.

Once you have both the slope and the y-intercept of a line, forming the equation is straightforward using the slope-intercept form. This form is represented by the formula:
\[y = mx + c\]
In this equation, \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope, and \(c\) is the y-intercept.
For the line that passes through the points \((2, 3)\) and \((6, -3)\), we have already determined that the slope \(m\) is \(-1.5\) and the y-intercept \(c\) is 6. Plugging these values into the slope-intercept form gives us:

• Replace \(m\) with \(-1.5\) and \(c\) with 6
• Write: \(y = -1.5x + 6\)

This equation \(y = -1.5x + 6\) provides a complete description of how \(y\) changes with \(x\). It tells us that with each increase of 1 in \(x\), \(y\) decreases by 1.5. This simple structure is widely used because it is easy to graph and interpret a line from these parameters.