91Ó°ÊÓ

The point-slope form of a linear equation is a valuable tool when you know a point on the line and the line's slope. This form is expressed as: \[ y - y_1 = m(x - x_1) \] where:

• \( (x_1, y_1) \) represents a known point on the line
• \( m \) is the slope of the line

To use this form effectively, you should already know the slope of the line or how to find it. But even before calculating the slope, just identify any point that the line passes through. Once you have a point and the slope, you can substitute these values into the point-slope form equation.
In our exercise, with the point (-8,0) and slope \( \frac{3}{7} \), we substitute into the formula to get:\[ y - 0 = \frac{3}{7}(x + 8) \].
From this equation, you could derive the line's entire equation, or simply use it to find another point on the line by plugging in values for \( x \).

The slope-intercept form is probably one of the most recognized forms of a linear equation. It's extremely useful for quickly visualizing where a line crosses the y-axis and its slope. The form is: \[ y = mx + b \] where:

• \( m \) is the slope
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

To convert a point-slope form equation to slope-intercept form, simply expand and simplify the expression. For example, in our exercise, starting with the point-slope form equation \( y - 0 = \frac{3}{7}(x + 8) \), we distribute and simplify to discover \( b \) as follows:

• Multiply: \( y = \frac{3}{7}x + \frac{24}{7} \)
• Here, \( \frac{3}{7} \) remains our slope, while \( \frac{24}{7} \) is our y-intercept.

This format lays out the slope and intercept clearly. Perfect for quick graphing, as you can directly plot the y-intercept and use the slope to mark other points.

Slope is a measure of how steep a line is. The slope refers to the change in the y-coordinate relative to the change in the x-coordinate between two points on a line.
Mathematically, it is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.

The formula essentially finds how much the y-coordinate changes as you move from one x-coordinate to another.
With our example, placing the points \( (-8, 0) \) and \( (-1, 3) \) into the slope formula, we get:\[ m = \frac{3 - 0}{-1 + 8} = \frac{3}{7} \].
This slope means that for every 7 units increase in x-direction, y increases by 3 units. Understanding this helps determine if the line rises or falls as you move along it.