The point-slope form of a line is a mathematical way to describe a line given its slope and a specific point that lies on the line. The formula for the point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here:
- [tex]\( m \)[/tex] represents the slope of the line.
- [tex]\( x_1 \)[/tex] and [tex]\( y_1 \)[/tex] are the coordinates of a specific point through which the line passes.
To understand why this form is valid, imagine we are given the slope [tex]\( m \)[/tex] and a point [tex]\((x_1, y_1)\)[/tex]. The slope [tex]\( m \)[/tex] of the line is defined as the change in [tex]\( y \)[/tex] over the change in [tex]\( x \)[/tex], or:
[tex]\[ m = \frac{y - y_1}{x - x_1} \][/tex]
To derive the point-slope form, we can start with the definition of the slope. If we multiply both sides of the slope equation by [tex]\((x - x_1)\)[/tex], we get:
[tex]\[ m(x - x_1) = y - y_1 \][/tex]
Rewriting it, we get the standard point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Thus, the point-slope form of a line correctly describes a linear relationship given a slope [tex]\( m \)[/tex] and a point [tex]\((x_1, y_1)\)[/tex]. This validates that the given statement is:
A. True
